Question

Find the general solution of the following differential equation: xy''+y'=x^2

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Jayne

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84 Answers

1. Identify the form of the differential equation

xy'' + y' = x^2

2. This is a linear differential equation. To solve it, we use the method of variation of constants.

3. First, solve the homogeneous equation:

xy'' + y' = 0

4. Simplify to a first-order equation by letting

y' = p

5. Then

y'' = p'

6. Substitute

x p' + p = 0

7. This is separable, so

p' = -\frac{p}{x}

8. Integrate

\ln|p| = -\ln|x| + C_1

9. Which means

p = \frac{C_1}{x}

10. Recall

y' = p = \frac{C_1}{x}

11. Integrate again

y = C_1\ln|x| + C_2

12. Now find the particular solution of the non-homogeneous differential equation using the method of undetermined coefficients.

13. Assume a particular solution in the form of

y_p = Ax^n

14. Find the correct power of \(x\), let's use \(y_p = Ax^3\)

15. Compute

y_p' = 3Ax^2

y_p'' = 6Ax

16. Substitute into the original equation:

x(6Ax) + 3Ax^2 = x^2

17. Simplify and solve for \(A\):

6Ax^2 + 3Ax^2 = x^2

9Ax^2 = x^2

A = \frac{1}{9}

18. Therefore,

y_p=\frac{x^3}{9}

19. Combine the general solution

y = y_h + y_p

20. Result:

y=\frac{x^3}{9}+C_1\ln|x|+C_2

2. This is a linear differential equation. To solve it, we use the method of variation of constants.

3. First, solve the homogeneous equation:

4. Simplify to a first-order equation by letting

5. Then

6. Substitute

7. This is separable, so

8. Integrate

9. Which means

10. Recall

11. Integrate again

12. Now find the particular solution of the non-homogeneous differential equation using the method of undetermined coefficients.

13. Assume a particular solution in the form of

14. Find the correct power of \(x\), let's use \(y_p = Ax^3\)

15. Compute

16. Substitute into the original equation:

17. Simplify and solve for \(A\):

18. Therefore,

19. Combine the general solution

20. Result:

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