I got y2 =et, W=tet, W1=-et, and W2=1+tFor my final answer I got c1(1+t)+c2et+et(1-t2/2 + t3/3). I'm not sure if it is right, but I can't find what, if anything I did wrong. ### Differential Equations: Solving ODEs**Problem Statement:**Solve the following ordinary differential equation (ODE):\[ t y'' - (1 + t)y' + y = t^2 e^t \]where one complementary solution is $ y_1 = 1 + t $.**Hint:** You may need to use Abel's theorem.**Detailed Explanation:**In this problem, you are given a second-order linear differential equation:\[ t y'' - (1 + t)y' + y = t^2 e^t \]and one complementary (particular) solution $ y_1 = 1 + t $.### Steps to Solve:1. **Substitute the given complementary solution** into the homogeneous part of the differential equation to verify it.2. **Apply Abel's Theorem** if necessary to find the second linearly independent solution.3. **Find the particular integral** of the differential equation using methods such as undetermined coefficients or variation of parameters.4. **Combine the complementary solution and the particular integral** to form the general solution.This problem provides a hint that Abel's theorem might be helpful. Abel's theorem is useful for solving linear differential equations and finding the Wronskian of the solutions.For a detailed step-by-step solution, please refer to the relevant sections of your differential equations textbook or consult with your instructor.**Keywords:** Ordinary Differential Equation, Complementary Solution, Abel's Theorem, Wronskian, Particular Integral.

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Answered: Solve the following ODE t y" – (1+t)y'…

Math

Advanced Math

Solve the following ODE t y" – (1+t)y' + y = t² e' where one complementary solution is y1 = 1+t. %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

I got y₂ =e^t, W=te^t, W₁=-e^t, and W₂=1+t

For my final answer I got c₁(1+t)+c₂e^t+e^t(1-t²/2 + t³/3).

I'm not sure if it is right, but I can't find what, if anything I did wrong.

### Differential Equations: Solving ODEs

**Problem Statement:**

Solve the following ordinary differential equation (ODE):

\[
t y'' - (1 + t)y' + y = t^2 e^t
\]

where one complementary solution is $ y_1 = 1 + t $.

**Hint:** You may need to use Abel's theorem.

**Detailed Explanation:**

In this problem, you are given a second-order linear differential equation:

\[
t y'' - (1 + t)y' + y = t^2 e^t
\]

and one complementary (particular) solution $ y_1 = 1 + t $.

### Steps to Solve:

1. **Substitute the given complementary solution** into the homogeneous part of the differential equation to verify it.
2. **Apply Abel's Theorem** if necessary to find the second linearly independent solution.
3. **Find the particular integral** of the differential equation using methods such as undetermined coefficients or variation of parameters.
4. **Combine the complementary solution and the particular integral** to form the general solution.

This problem provides a hint that Abel's theorem might be helpful. Abel's theorem is useful for solving linear differential equations and finding the Wronskian of the solutions.

For a detailed step-by-step solution, please refer to the relevant sections of your differential equations textbook or consult with your instructor.

**Keywords:** Ordinary Differential Equation, Complementary Solution, Abel's Theorem, Wronskian, Particular Integral.

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