2.4E: First Order Linear Differential Equations (Exercises)

 In Exercises 1-17,  find the general solution of the differential equation using the method of integrating factor. Include the largest interval over which the solution is defined.

1. \(y'+3x^2y=0\)

2. \(\dfrac{dx}{dt}+x=e^{3t}\)

3. \(y'+2xy=x^3\)

4. \(\cos x \dfrac{dy}{dx}+(\sin x)y=1\)

5. \(x^2y'+xy=1\)

6. \(\dfrac{dr}{d\theta}+r\sec\theta=\cos\theta\)

7. \( {y'+\left({1\over x}- 1\right)y=-{2\over x}}\)

8. \(y'+2xy=xe^{-x^2}\)

9. \( {y'+{2x\over1+x^2}y={e^{-x}\over1+x^2}}\)

10. \( {y'+{1\over x}y={7\over x^2}+3}\)

11. \( {y'+{4\over x-1}y = {1\over (x-1)^5}+{\sin x\over (x-1)^4}}\)

12. \(xy'+(1+2x^2)y=x^3e^{-x^2}\)

13. \( {xy'+2y={2\over x^2}+1}\)

14. \(y'+(\tan x)y=\cos x\)

15. \( {(1+x)y'+2y={\sin x \over 1 + x}}\)

16. \((x-2)(x-1)y'-(4x-3)y=(x-2)^3\)

17. \(y'+(2\sin x\cos x) y=e^{-\sin^2x}\)

18. \(x^2y'+3xy=e^x\)

 In Exercises 19-33, solve the initial value problem using the method of integrating factor. Include the largest interval over which the solution is defined.

19. \( {y'+\left({1+x\over x}\right)y=0,\quad y(1)=1}\)

20. \( {xy'+\left(1+{1\over\ln x}\right)y=0,\quad y(e)=1}\)

21. \( {xy'+(1+ x\cot x)y=0,\quad y\left({\pi\over 2} \right)=2}\)

22. \( {y'-\left({2x\over 1+x^2}\right)y=0,\quad y(0)=2}\)

23. \(y'+7y=e^{3x},\quad y(0)=0\)

24. \( {(1+x^2)y'+4xy={2\over 1+x^2},\quad y(0)=1}\)

25. \( {xy'+3y={2\over x(1+x^2)},\quad y(-1)=0}\)

26. \( {y'+ (\cot x)y=\cos x,\quad y\left({\pi\over 2}\right)=1}\)

27. \( {y'+{1\over x}y={2\over x^2}+1,\quad y(-1)=0}\)

28. \( {(x-1)y'+3y={1\over (x-1)^3} + {\sin x\over (x-1)^2},\quad y(0)=1}\)

29. \(xy'+2y=8x^2,\quad y(1)=3\)

30. \(xy'-2y=-x^2,\quad y(1)=1\)

31. \(y'+2xy=x,\quad y(0)=3\)

32. \( {(x-1)y'+3y={1+(x-1)\sec^2x\over (x-1)^3},\quad y(0)=-1}\)

33. \( {(x+2)y'+4y={1+2x^2\over x(x+2)^3},\quad y(-1)=2}\)

 

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