1. If T:R2→R2 is defined by T(x,y)=(3x−1,x−4y), deteine whether T is linear 2. Use the definition of linearity to show that if S:Fn→Fm and T:Fm→Fp are both linear, than so is T∘S. Make sure to justify each step.
Answers
T∘S is a linear transformation.
1. If T:R2→R2 is defined by T(x,y)=(3x−1,x−4y), determine whether T is linear.To show that T is linear, we need to prove that T satisfies the two properties of linearity. i.e. T satisfies additivity and homogeneity. Let x1,y1,x2,y2 be arbitrary vectors in R2 and let a be an arbitrary scalar in R. Then,T((x1,y1)+(x2,y2))=T(x1+x2,y1+y2)=(3(x1+x2)−1,(x1+x2)−4(y1+y2))= (3x1−1,x1−4y1)+(3x2−1,x2−4y2)=T(x1,y1)+T(x2,y2)and T(a(x1,y1))=T(ax1,ay1)=(3(ax1)−1,(ax1)−4(ay1))=a(3x1−1,x1−4y1)=aT(x1,y1)Hence, T is a linear transformation.2. Use the definition of linearity to show that if S:Fn→Fm and T:Fm→Fp are both linear, then so is T∘S. Make sure to justify each step.Given, S:Fn→Fm and T:Fm→Fp are both linear,We need to prove that T∘S is also a linear transformation. Let x,y be arbitrary vectors in Fn and a be an arbitrary scalar in F. Then,(T∘S)(x+y)=T(S(x+y))=T(S(x)+S(y))=T∘S(x)+T∘S(y)and (T∘S)(ax)=T(S(ax))=T(aS(x))=aT∘S(x)Hence, T∘S is a linear transformation.
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Related Questions
What is the slope of the line that passes through the points (-5, 8) and
(-5, 4)? Write your answer in simplest form
Answers
Answer:
Step-by-step explanation:
I'm pretty sure its undefined :))
What is QR ? Enter your answer in the box. Units The figure shows what appears to be obtuse triangle Q R S with obtuse angle R. Point T is on side Q R. Single tick marks pass through segments Q T and T R. Point U is on side R S. Double tick marks pass through segments R U and U S. Point V is on side S Q. Triple tick marks pass through segments S V and V Q. Segment T V is drawn and has length 5. 4. Segment U V is drawn and has length 6
Answers
QR refers to a Quick Response code that is similar to a barcode that can be scanned with a smartphone to read the information it holds.
The Quick Response (QR) code is a type of two-dimensional (2D) matrix barcode that consists of black and white square dots arranged in a square grid on a white background. QR codes are frequently used to encode URLs or other information that can be scanned and read by a smartphone. They are used in a variety of applications, including advertising, product packaging, and business cards.To explain the given figure, an obtuse triangle QRS is given, which has an obtuse angle R. Point T is on side QR, Point U is on side RS, and Point V is on side SQ. Single tick marks pass through segments QT and TR.Double tick marks pass through segments RU and US.Triple tick marks pass through segments SV and VQ. Segment TV is drawn, which has a length of 5 units, and segment UV is drawn, which has a length of 6 units.For such more questions on QR
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Find an autonomous differential equation with all of the following properties:
equilibrium solutions at y=0 and y=3,
y' > 0 for 0 y' < 0 for -inf < y < 0 and 3 < y < inf
dy/dx =
Answers
y(y - 3) = 0
Expanding the equation:
y^2 - 3y = 0
Now, let's consider the signs of y' in different intervals:
1. For 0 < y < 3, we want y' to be positive. We can introduce a factor of y on the right-hand side of the equation to ensure this:
y' = ky(y - 3)
2. For y < 0 and y > 3, we want y' to be negative. We can introduce a negative factor of y on the right-hand side to achieve this:
y' = ky(y - 3)(y - 0)
Where k is a constant that determines the rate of change.
Combining the conditions, we can write the autonomous differential equation with the given properties as:
y' = ky(y - 3)(y - 0)
This equation has equilibrium solutions at y=0 and y=3, and satisfies the conditions y' > 0 for 0 < y < 3, and y' < 0 for y < 0 and y > 3.
all the three terms on the right-hand side are positive and hence dy/dx is negative. Thus, this satisfies all the properties given. Therefore, the required autonomous differential equation is:dy/dx = a (y - 3) (y) (y - b).
We can obtain the autonomous differential equation having all of the given properties as shown below:First of all, let's determine the equilibrium solutions:dy/dx = 0 at y = 0 and y = 3y' > 0 for 0 < y < 3For -∞ < y < 0 and 3 < y < ∞, dy/dx < 0This means y = 0 and y = 3 are stable equilibrium solutions. Let's take two constants a and b.a > 0, b > 0 (these are constants)An autonomous differential equation should have the following form:dy/dx = f(y)To get the desired properties, we can write the differential equation as shown below:dy/dx = a (y - 3) (y) (y - b)If y < 0, y - 3 < 0, y - b < 0, and y > b. Therefore, all the three terms on the right-hand side are negative and hence dy/dx is positive.If 0 < y < 3, y - 3 < 0, y - b < 0, and y > b. Therefore, all the three terms on the right-hand side are negative and hence dy/dx is positive.If y > 3, y - 3 > 0, y - b > 0, and y > b. Therefore, all the three terms on the right-hand side are positive and hence dy/dx is negative. Thus, this satisfies all the properties given. Therefore, the required autonomous differential equation is:dy/dx = a (y - 3) (y) (y - b).
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How do you write coordinates in XY?
Answers
We use the format (x, y) to write the coordinates in format XY. Coordinates XY is a coordinate that use in Cartesian coordinate.
What is Cartesian coordinate?
Cartesian coordinate is system to draw a specific location of an unique point in the numerical condition that oriented in the two constant line where the vertical line called ordinate or Y and the horizontal line called abscess or X. The Cartesian coordinate commonly use in architecture or a design of a certain object that has a certain dilate with its original plan. This design later is use to control the construction process of the original object.
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About how many times greater was change in price per gallon in 2007 than 2000? Show your work or explain how u determind your answer.
Answers
The required, in 2007 the price per gallon was 7b more than the price of a gallon of fuel in the year 2000. Where b is the inflation factor.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Let 'a' be the cost per gallon of fuel in the year 2000, and 'b' be the inflation rate per year. If the rate of inflation is constant then
After 7 year inflation = 7b
Cost of fuel in 2007 = a + 7b
Now,
according to the question
Change in cost of fuel
= cost in 2007 - cost in 2000
= a + 7b - a
= 7b
Thus, the required, in 2007 the price per gallon was 7b more than the price of a gallon of fuel in the year 2000. Where b is the inflation factor.
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ID=48
Two disjoint sets {y, p, z, x} and {r, t} are given, where minimum one of a set is the representative of that set. Determine UNION(Find(x), Find(t)). How can you check x and y are in the same set using Find operation? Here, x=last two digits of your student id+7, y=x+3, z=x+y, p=y+z, r=x+2, t=900.
Answers
The main answer is: UNION(Find(x), Find(t)) = UNION(Find(48), Find(900)) = UNION({x}, {t}) = {x, t}.
To determine UNION(Find(x), Find(t)), we need to find the representatives (minimum elements) of the sets that x and t belong to. From the given information, we know that x = 48 and t = 900. Since each element is its own representative, Find(x) = {x} and Find(t) = {t}. Taking the union of these two sets gives us UNION(Find(x), Find(t)) = {x} ∪ {t} = {x, t}. Therefore, the answer is {x, t}.
To check if x and y are in the same set using the Find operation, we need to find their respective representatives and see if they match. In this case, y = x + 3, and z = x + y. Since we know the value of x (48), we can calculate y = 48 + 3 = 51 and z = 48 + 51 = 99. Now, to find the representatives of x and y, we use the Find operation. Find(x) = {x} and Find(y) = {y}. If the representatives of x and y are the same, it means x and y are in the same set. In this case, since Find(x) = {x} and Find(y) = {y}, and both sets contain only a single element, we can conclude that x and y are in the same set.
Therefore, by comparing the representatives obtained from the Find operation, we can determine if x and y are in the same set.
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The main answer is: UNION(Find(x), Find(t)) = UNION(Find(48), Find(900)) = UNION({x}, {t}) = {x, t}.
To determine UNION(Find(x), Find(t)), we need to find the representatives (minimum elements) of the sets that x and t belong to. From the given information, we know that x = 48 and t = 900. Since each element is its own representative, Find(x) = {x} and Find(t) = {t}. Taking the union of these two sets gives us UNION(Find(x), Find(t)) = {x} ∪ {t} = {x, t}. Therefore, the answer is {x, t}.
To check if x and y are in the same set using the Find operation, we need to find their respective representatives and see if they match. In this case, y = x + 3, and z = x + y. Since we know the value of x (48), we can calculate y = 48 + 3 = 51 and z = 48 + 51 = 99. Now, to find the representatives of x and y, we use the Find operation. Find(x) = {x} and Find(y) = {y}. If the representatives of x and y are the same, it means x and y are in the same set. In this case, since Find(x) = {x} and Find(y) = {y}, and both sets contain only a single element, we can conclude that x and y are in the same set.
Therefore, by comparing the representatives obtained from the Find operation, we can determine if x and y are in the same set.
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Yorktown Savings Bank, in reviewing its credit card customers, finds that of 30 percent (or a total of 7,665 customers) of those customers who scored 40 points or less on its credit-scoring system were in default of their accounts, resulting in a total loss of their account balances. This group of bad credit card loans averaged $6,200 in size per customer account. When examining its successful credit accounts. Yorktown finds that 12 percent of its good customers (or a total of 3,066 customers) scored 40 points or less on the bank's scoring system. These low-scoring but good accounts generated about $1,000 in revenue per account. If Yorktown's credit card division follows the decision rule of granting credit cards only to those customers scoring more than 40 points, about how much can the bank expect to save in net losses? Please input your answer in the xx,xxx,xxx format and round to the nearest whole dollar. Enter $52.849,023 as 52.849,023.
Answers
Yorktown Savings Bank can expect to save approximately $1,444,400 in net losses by following the decision rule of granting credit cards only to customers scoring more than 40 points.
To calculate the expected savings in net losses, we need to determine the number of bad accounts that would be avoided by applying the decision rule and multiply it by the average account balance of those accounts.
The total number of customers who scored 40 points or less and resulted in a loss is 7,665, representing 30% of all customers. On the other hand, 12% of good customers, which amounts to 3,066 customers, also scored 40 points or less.
By applying the decision rule of granting credit cards only to customers scoring more than 40 points, we can estimate the number of bad accounts that would be avoided as 30% of the customers who scored 40 points or less, i.e., 0.3 * 3,066 = 920.
The average account balance of the bad accounts is $6,200. Multiplying this by the number of accounts avoided, we find that the expected savings in net losses would be approximately $1,444,400.
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Plea help me get the answer
Answers
WILL MARK BRAINLIEST!
arrange -2, 1.8, -1/3, -1/4, √5 at descending order
Answers
Answer:
√5, 1.8, -1/4,-1/3,-2
Step-by-step explanation:
Please put brainliest if this helps
Answer:
sqrt5, 1.8, -1/4, -1/3 and -2
Step-by-step explanation:
I. Find the true value of variables
-2 = -2
1.8 = 1.8
-1/3 = -0.333...
-1/4 = -0.25
sqrt5 = 2.236
II. Descending is most to least so
= 2.236, 1.8, -0.25, -0.333, -2
or sqrt5, 1.8, -1/4, -1/3 and -2
For -0.x, if x is near 0 will be the greatest value,
and +0.x if x is far from 0 will be the greatest value
So Descending is sqrt5, 1.8, -1/4, -1/3 and -2
What is the maximum of thisgraph over the interval [-4, 4]?
Answers
The maximum of a graph is found on the y axis. Considering the given interval, the point on the y axis is 4. Thus, the maximum of the graph over the interval is 4.
What is the surface area and volume of a pentagonal prism?
Answers
The surface area and volume of a pentagonal prism is 5/2 × a² × √(5 + 2√5) + 5ab and (1/4) × (5 + 2√5) × a² × h respectively. We can find the solution in the following manner.
A pentagonal prism is a three-dimensional geometric shape that consists of two parallel pentagons as the top and bottom faces, and five rectangular faces connecting them.
To find the surface area and volume of a pentagonal prism, we need to know its height, the length of the sides of the pentagon, and the length of the rectangular faces.
Let's denote the height of the pentagonal prism as "h", the side length of the pentagon as "a", and the length of the rectangular face as "b".
Surface Area of a Pentagonal Prism:
The surface area of a pentagonal prism is the sum of the areas of its faces. There are two pentagonal faces and five rectangular faces in a pentagonal prism.
Area of each pentagonal face = 5/4 × a² × √(5 + 2√5)
Area of each rectangular face = a × b
Total surface area = 2 × Area of pentagonal face + 5 × Area of rectangular face
= 5/2 × a² × √(5 + 2√5) + 5ab
Volume of a Pentagonal Prism:
The volume of a pentagonal prism is given by the formula:
Volume = (1/4) × (5 + 2√5) × a² × h
Therefore, the surface area and volume of a pentagonal prism can be calculated using the above formulas, given the values of a, b, and h.
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The surface area and volume of a pentagonal prism is sum of the areas of its faces, and (1/4) × (5 + 2√5) × a² × h.
A pentagonal prism is a three-dimensional geometric shape that consists of two parallel pentagons as the top and bottom faces, and five rectangular faces connecting them.
Surface Area of a Pentagonal Prism:
The surface area of a pentagonal prism is the sum of the areas of its faces. There are two pentagonal faces and five rectangular faces in a pentagonal prism.
Total surface area = 2 × Area of pentagonal face + 5 × Area of rectangular face
Volume of a Pentagonal Prism:
The volume of a pentagonal prism is given by the formula:
Volume = (1/4) × (5 + 2√5) × a² × h
Therefore, the surface area and volume of a pentagonal prism can be calculated using the above formulas, given the values of a, b, and h.
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Can some one help please
Answers
Answer: \(\fbox{C} \: \: \: \dfrac{4 \times 10^{4} }{3 \times 10^{4} }\)
Step-by-step explanation:
36,070 = 3.607 · 10⁴ ≈ 4 · 10⁴
29,029 = 2.9029 · 10⁴ ≈ 3 · 10⁴
Let's how manytimes greater the distance to the bottom of Challenger Deep is than the distance to the top of Mount Everest:
\(\dfrac{4 \times 10^{4} }{3 \times 10^{4} }\)
Answer:
C
Step-by-step explanation:
First, the question is asking how many times greater the distance to the Challenger Deep is compared to Mount Everest. This means that the Challenger Deep will be on top, with the distance to Mount Everest on the bottom.
Next, we can see that in our answers, we are only given one significant digit. This means that we have to round our original numbers to the nearest significant digit and work from there. For Challenger Deep, 36,070 must be rounded to the nearest ten thousandth as it is in the ten thousands. Therefore, it must be rounded to 40,000 or 30,000. As 36,070 is closer to 40,000 than 30,000 (we can tell this because the second number, 6, is greater than 5, so we round up), we can represent this as 40,000. For Mount Everest, we can round 29,029 to either 20,000 or 30,000. 30,000 is the appropriate choice here because 9, the second number, is greater than 5.
Therefore, our ratio is now 40,000 / 30,000 . Our answers are written in scientific notation, so we must convert our numbers to those values.
Let's start with 40,000 . Our one significant digit is 4, so we can write this as 4 * something. For each number in 10 to the power of x (x is a placeholder value), we add x amount of zeros. For example, for 4 * 10², we add 2 zeros, making it 400. There are 4 zeros here, so to write this in scientific notation, we have 4 * 10^4
Similarly, for 30,000 , we can write this as 3 * 10^4 as there are 4 zeros.
Our ratio is thus 40,000 = 4*10^4 , which is the Challenger Deep value rounded, over 30,000 = 3 * 10^4, which is the Mount Everest value rounded. We can write this as \(\frac{4*10^{4}}{3*10^{4}}\), or C
1. What are the 3 conditions for a function to be continuous at xa? 2. the below. Discuss the continuity of function defined by graph 3. Does the functionf(x) = { ***
Answers
The three conditions for a function to be continuous at a point x=a are:
a) The function is defined at x=a.
b) The limit of the function as x approaches a exists.
c) The limit of the function as x approaches a is equal to the value of the function at x=a.
The continuity of a function can be analyzed by observing its graph. However, as the graph is not provided, a specific discussion about its continuity cannot be made without further information. It is necessary to examine the behavior of the function around the point in question and determine if the three conditions for continuity are satisfied.
The function f(x) = { *** is not defined in the question. In order to discuss its continuity, the function needs to be provided or described. Without the specific form of the function, it is impossible to analyze its continuity. Different functions can exhibit different behaviors with respect to continuity, so additional information is required to determine whether or not the function is continuous at a particular point or interval.
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What is the vertex?
f(x) = -6x² + 12x + 18
Answers
What is the simplified expression for Negative 3 c d minus d (2 c minus 4) minus 4 d? Negative 5 c d c d minus 8 d 5 c d minus 8 d Negative 5 c d minus 8 d
Answers
Question
What is the simplified expression for Negative 3 c d minus d (2 c minus 4) minus 4 d?
a) Negative 5 c d
b) c d minus 8 d
c) 5 c d minus 8 d
d) Negative 5 c d minus 8 d
Answer:
a) Negative 5cd or -5cd
Step-by-step explanation:
The question asks:
What is the simplified expression for Negative 3 c d minus d (2 c minus 4) minus 4 d?
This is expressed mathematically as:
-3cd - d(2c - 4) - 4d
= -3cd -2cd + 4d - 4d
= -5cd + 0
= -5cd
The answer is negative 5cd(in words) or - 5cd
A customer borrows $450 at 5% interest compounded annually
Answers
when a class interval is expressed as 100 up to 200, _________________________.
Answers
When a class interval is expressed as 100 up to 200, it means that the data is grouped into intervals or ranges, and the first interval starts at 100 while the last interval ends at 200.
When dealing with large sets of data, it is often more convenient to group the data into intervals or classes. Each interval is a range of values, and the frequency of data falling within that range is recorded. The class interval "100 up to 200" means that the first interval starts at 100, and the range continues up to but does not include 200.
This means that the first interval will include all values greater than or equal to 100 and less than 200. The exact size of the interval (i.e., the width) is not specified in this expression, so it could be any value that covers the range between 100 and 200.
For example, the interval could be 100-199, 100-199.99, or any other width that covers the specified range.
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Am a stands beside the road and counts cars and bikes. If she counts 250 wheels in total, how many cars and wheels are there
Answers
Am a stands beside the road and counts cars and bikes. If she counts 250 wheels in total, Anna has seen 50 cars and 25 bikes.
To calculate this, use linear equation.
Number of cars and bikes
Let C represent Cars and B represents Bike
A car has 4 wheels and a bike has 2 wheels;
So,
\(4C+2B=250\)
Divide through by 2
\(\frac{4C+2B}{2} =\frac{250}{2}\)
\(2C+B=125\) ---------equation 1
The ratio of wheels of cars to wheels of bike is 1:2
Meaning that
1C = 2B
So, C = 2B
Substitute 2B for C in equation 1
\(2(2B)+B=125\\4B+B=125\\5B=125\)
Divide both sides by 5
\(\frac{5B}{5} =\frac{125}{5} \\B=\frac{125}{5} \\B=25\)
Recall that C = 2B
So,
\(C= 2*25\\C=50\)
Hence, Am a has seen 50 cars and 25 bikes.
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When x=3, the value of x^2-2 is?
Answers
Answer:
7
Step-by-step explanation:
First you would square 3 which is nine. Then subrtact 2. The final answer is 7 .
y=x^{4}
input/output pairs for the function
Answers
The input/output pairs for the function are (0, 0), (1, 1), (2, 16) and (3, 81)
How to determine the input/output pairs for the functionFrom the question, we have the following parametes that can be used in our computation:
y = x^4
Set x = values from 0 to 3
So, we have the following representation
y = 0^4 = 0
y = 1^4 = 1
y = 2^4 = 16
y = 3^4 = 81
When represented as ordered pairs, we have
(0, 0), (1, 1), (2, 16) and (3, 81)
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Find the area of the circle. Round your answer to the nearest whole number, if necessary.
A circular object with a radius of 10 inches.
area: about
in.2
Answers
i need help solving the first problem
Answers
Answer:
f(x), g(x)10, 0, UNDEFINEDStep-by-step explanation:
You want to know which functions have 5 in their domain, and what the function value is at x=5.
f(x) = x² -3xg(x) = (x -5)/xh(x) = √(x -10)ValuesIt is more convenient to do the second part first, as that tells you the answer to the first part:
f(5) = 5² -3·5 = 25 -15 = 10 . . . . 5 is in the domain
g(5) = (5 -5)/5 = 0/5 = 0 . . . . 5 is in the domain
h(5) = √(5 -10) = √(-5) . . . . UNDEFINED; 5 is not in the domain
which line has negative slope
Answers
Answer:
the answer is C
Step-by-step explanation:
The negative graph has
m= -ve ( slop is negative value)It's slop is downward ( from Qll to QlV )So the equetion for the negative slop is y= -m + b
from the given 3 equetion C fulfill all the requirment .
So the answer is C
I'm 17 and a mom to 3 11-year olds. I was wondering if this goes together because my daughter Angelynn does dance and has an Acro solo at the next competition.
Answers
Answer:
Yah i think its nice but I think its too much makeup for a kid though but its still godg for a dancer:)
Step-by-step explanation:
Answer:
yeah it goes together.
A clinic provides a program to help their clients lose weight and asks a consumer agency to investigate the effectiveness of the program. The agency takes a sample of 15 people, weighing each person in the sample before the program begins and 3 months later.
Which hypothesis test methods would be appropriate for this data set? Select all that apply.
A. Independent t test
B. Paired t test
C. ANOVA
D. Nonparametric paired test
Answers
The appropriate hypothesis test methods for this data set are:
B. Paired t-test
D. Nonparametric paired test
We have,
Since the agency is measuring the weight of the same individuals before and after the program, a paired test is suitable.
The paired t-test is appropriate if the data follows a normal distribution and the differences between the paired observations are approximately normally distributed.
If the assumptions for the paired t-test are not met, a nonparametric paired test (such as the Wilcoxon signed-rank test) can be used as an alternative.
ANOVA and independent t-tests are not appropriate for this data set since they involve comparing independent groups, which is not the case here.
Thus,
The appropriate hypothesis test methods for this data set are:
B. Paired t-test
D. Nonparametric paired test
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The circumference of a circle is 6.28 kilometers. What is the circle's radius?
Use 3.14 for Pi.
Answers
Answer:
The radius is 1 km.
Step-by-step explanation:
c =2\(\pi\)r
6.28 = 2(3.14)r
6.28 = 6.28r Divide both sides by 6.28
1 = r
The radius is 1 km.
Helping in the name of Jesus.
We know that the formula for circumference is:
C = 2πr
where C is the circumference and r is the radius.
Substituting the given value for C:
6.28 km = 2πr
Dividing both sides by 2π:
r = 6.28 km / 2π
r ≈ 1 km
Therefore, the radius of the circle is approximately 1 kilometer.
tim and david are baking pies for a fundraiser at their school. the divide each 9-inch diameter pie into 6 equal slices. what is the area of a piece of pie in square inches?
Answers
Therefore, the area of a piece of pie is 10.60 square inches.
To find the area of a pie with a 9-inch diameter, we can use the formula for the area of a circle:
A = πr²
where r is the radius of the circle. Since the diameter of the pie is 9 inches, the radius is half of that, or 4.5 inches. So we can plug this value into the formula to get:
A = π(4.5)²
Simplifying, we get:
A = π(20.25)
A = 63.62 square inches (rounded to two decimal places)
This is the total area of the pie.
Since the pie is divided into 6 equal slices, we can divide the total area by 6 to get the area of each slice. So we can calculate the area of each slice as:
63.62 / 6 = 10.60 square inches (rounded to two decimal places)
Therefore, each slice of the pie has an area of approximately 10.60 square inches.
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HELP ME PLEASE!!!!!!!!
Answers
(2). The reason i picked yes is because very amount of money is $2.50 away from each other so this is a linear relationship.
Simplify using the Distributive Property.
7 × 48
A. (7 × 50) − (7 × 2) = 350 – 14 = 336
B. (7 × 40) + 8 = 288
C. (7 × 50) – 2 = 348
D. (7 × 40) + (7 x 8) + 7 = 343
Answers
Answer:
B
Step-by-step explanation:
write the following as a system of first-order equations (t 1)2 d 3 y dt3 d 2 y dt2 2 dy dt 6y(t)
Answers
The system of first-order equations that is equivalent to the given second-order differential equation is dy/dt = z, dz/dt = w, and dw/dt = (-3z - 2w - 6y)/t².
To write the given second-order differential equation as a system of first-order equations, we need to introduce new variables.
Let z = dy/dt. Then, we can rewrite the given equation as
d³y/dt³ = dz/dt
d²y/dt² = dz/dt = z
Substituting these expressions into the original equation, we get
(t²) (d³y/dt³) + 3(d²y/dt²) + 2(dy/dt) + 6y = t² (dz/dt) + 3z + 2(dy/dt) + 6y
Simplifying and grouping the terms, we obtain
d/dt [y, z, w] = [z, w, (-3z - 2w - 6y)/t²]
where w = dt/dt = 1.
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