Exercise 5.1 (Ordinary Differential Equations)

EXERCISE 5.1

 

I. (a) R.H.S. = 0 Type

Solve the following:

1. d²x /dt – 8dx/dt +16x = 0  

[Ans. x = e^4t [C₁t + C₂]

 

2. (D² + D + 1) y = 0 

[Ans. y = e^-x/2 [C₁ cos √3/2 x + C₂ sin √3/2 x]]

 

3. (3D₂ + D - 14) y = 0 

[Ans. y = C₁e^-7/3x + C₂ e^2x]

 

I. (b) R.H.S. = e^ax type

Solve the following:

1. (D² - 6D + 9) y = 6e^3x 

[ Ans. (Ax + B)e^3x + 3x²e^3x]

 

2. (D² - 2D + 1) y = 5e^3x + cosh 2x

Hint : R.H.S. = 5e^3x + e^2x + e^-2x /

[Ans. y = (Ax+B) e^x + 5e^2x/4 + e^2x/2 + e^-2x/18 ]

 

3. (3D2 + D − 14)y = sinhx 

[Ans. C₁ e^-7/3x + C₂e^2x – e^x/20 + e^-x/24 ]

 

4. (D² − 1)y = sinh  

[Ans. y = C₁e^x + C₂ e^-x + ½ x cosh x ]

 

5. (D² + 6D + 5) y = e^2x   

[Ans. Ae^-5x + Be^-x + 1/21 e^2x]

 

6. (4D² - 4D + 1) y = 4 

[ Ans. y = (A + Bx ) e ^{½ x} + 4 ]

 

7. (D³ + 2D² + D) y = e^2x  

[ Ans . A + ( B + Cx )e^-x +  1/18 e^2x ]

 

8.(D³ - 12D + 6) y = (e^x + e^−2x) ² 

[Ans. y = Ae^-4x + ( B + Cx )e^2x + x²/12 e^2x + 2/27 e^-x + x/36 e^-4x ]

 

9. (D² - 4) y = 6 e^5x  

[  Ans. y = Ae^2x + Be^-2x + 2/7 e^5x ]

 

10. (D² + 2D + 1) у = л 

[ Ans. y = (Ax + B)e^-x + π

 

11. (D² + 2D + 5) y = 15 

 

 

12. (D2 + 7D +12) y = e2x +6 

[ Ans. y = Ae^-3x + Be^-4x + 1/30 e^2x + ½ ]

 

13. (D² - 3D - 4) y = e^3x + e^-x 

[ Ans. y = Ae^4x + Be^-x – e^3x/4 – xe^-x/5 ]

 

14. (3D² + D -14) y = 5 + 13 e^2x 

[ Ans. y = 1e^2x + Be^{-7/3 x} – 5/14 + xe^2x ]

 

15. (D² + 2D + 1) y = e^-x + 3 

 [Ans. y = (Ax + B) e^-x + x²/2 e^-x + 3 ]

 

16. (D² + 5D + 6) y = e^-7x  sinh 3x 

 

17. (D² + 2D + 1) y = cosh 2x 

 

18. (D³ - 3D² + 4D - 2) y = sinh²x 

[Ans. y = C₁e^x + (C₂ cos x + C₃ sin x) e^x + 1/18 e^2x – 1/120 e^-2x + ¼ ]

 

19. (D² - 2D + 1) y = (e^x + 1)²   

[Ans. y = (Ax + B) e^x + e^2x + 1+ x² e^x]

 

I. (c) R.H.S. = cos ax (or) sin ax

Solve the following:

1.d²y/dx² + 4y = sin 2x.  

[Ans. y = A cos 2x + B sin 2x – x/4 cos 2x ]

 

2. (D²+16) y =cos 4x 

[Ans. y = A cos 4x + B sin 4x + x/8 sin 4x]

 

3. (D² + D + 1)y =  sin 2x 

 

4. (D² - 4D + 3) y = sin 3x cos 2x

Hint: R.H.S. = [sin 5x + sin x]

 

5. (D² - 3D + 2) y = cos 3x cos 2x

[Ans. y =C₁e^2x + C₂e^2x + 1/20 [  cosx − 3 sin x] – 1/1508 [ 23 cos 5x + 15 sin 5x]]

 

6. (D²- 5D + 6) =  sin x sin 2x 

[Ans. y = C₁e^2x + C₂ e^3x + 1/20[ cos x - sin x] + 1/156 (cos 3x + 5 sin 3x)]

 

7. (D² + 4)2 y = cos 2x

[Ans. y = (C₁ + C₂x) cos 2x + (C₃ + C₄x) sin 2x – 1/32 x²       cos 2x]

 

8. (D² + 3D + 2) y = sin 3x cos 2x 

I. (d) R.H.S.= e^ax + [cos ax or sin ax]

Solve the following:

 

1. (D² + 1) y = cos² x

Hint: R.H.S. = 1 + cos 2x/2 = ½ e^0x + ½ cos 2x

 

2. d²y /dx² – 4 dy/dx + 3y = e^2x + cos x

 

 

3. d²y/dx² + 4y = e^x + sin 2x 

[ Ans.  y = C₁cos 2x + C₂ sin 2x + 1/5 e^x – x/4 cos 2x ]

 

4. (D² + 1) y = sin²x 

[Ans. y = C₁ cos x + C₂ sin x + ½ + 1/ cos 2x ]

 

5. (D²+16) y = e^-3x + cos 4x 

[Ans. y = C₁ cos 4x + C₂ sin 4x+ 1/25e_-3x + 1/8 x sin 4x]

 

6. (D² - 3D + 2) y = 2 cos (2x+3)+ 2e^x [A.U A/M. 2005]

[Ans. y = C₁e^x + C₂ e^2x – 1/10 [3 sin (2x + 3) + cos (2x+3)] − 2 xe^x]

 

7. (D² - 4D -5) y = cos x + e^-x  

[Ans. y = A e^5x + Be ̄^-x – 1/26  (3 cos x + 2 sinx) – x/6 e^-x]

 

8. (D³ + 2D² + D) y = e^2x + sin x 

[Ans. y = A + (B + Cx) e^-x + 1/18 e^2x -  ½ sin x]

 

9. (D³ + D² – D - 1) y = cos 2x + 7 

[Ans. y = A e^x + (B + Cx) e^-x – 1/25 [2 sin 2x + cos 2x] - 7]

 

10. (D³ + 3D² + 3D + 1) y = 5 + cos 2x 

[Ans. y = (4 + Bx + Cx²) e ^-x +5 – 1/125 [2 sin 2x + 11 cos 2x]]

 

11. (D² - 3D + 2) y = 2 cos (2x+3)+2e^x [Ans. y = Ae^x + Be^2x – 1/10 [cos (2x+3)] -3/10 [sin(2x+3)] − 2 xe^x]

 

12. (D² + 6D+8) y = 11 cos²x 

 

13. (D² + 9) y = 11 cos³x  

[Ans. y = (A cos 3x + B sin 3x) + 11/24 x sin 3x + 33/32 cos x]

 

I. (e) R.H.S = xⁿ type.

Solve the following:

1. (D² + 3D + 2) y = x² 

[Ans. y = C₁e^-2x + C₂ e ̄^-x + 1/2 (x2 - 3x + 2)]

 

2. d²y/dx² + 2 dy/dx + 3y = 5x² 

 

3. (D³ - D²- D + 1)y = 1+x²  

[Ans. y = C₁e^-x+ (C₂+ C₃x) e^x + x² + 2x + 5]

 

4. (D³ + 3D² + 2D) y = x² + 1  

 

5. ( D³ + D)y = x² + 2x + 4 [A.U N/D 2000] 

[ Ans. y = C₁ + C₂ e^-x + 1/3 x³ + 4x ]

 

6. (D² + 5D + 4) y = x² + 7x + 9 

 

7. D² (D² + 4) y = 96x² 

[Ans. y = (C₁x + C₂) + C₃ cos 2x + C₄ sin 2x + 2x⁴ - 6x²]

 

8. (D² – 5D + 6)y = x² + 3 

 

9. (D³ - D² - 6D) y = x² + 1

[Ans. y = C₁+ C₂e^-2x + C₃ e^3x – 1/6 [ x³/3 – x²/6 + 25/18 x]].

 

10. (D³ + 2D² - D - 2) y = 1-4x³ 

[Ans. y = A e^x + Be^-x + Ce^-2x + 2x³ - 3x² + 15/2 ]

 

11. (D⁴ + 8D² + 16) y = 16x + 10 

[Ans. y = (C₁+ C₂x) cos 2x + (C₃ + C₄x) sin 2x + 1/16 (16x + 10)]

 

12. D² (D² + 4) y = 320 (x³ + 2x²) 

[Ans. y = C₁ + C₂x + C₃ cos 2x + C₄ sin 2x + 4x⁵ + 40/3 x⁴ +- 20x³ - 40x²]

 

I. (f) R.H.S.= e^ax X type

Solve the following:

1.(D² - 2D + 2) y = e^x sin x 

[Ans. y=e^x [A cosx + B sinx] – ½ x cos x e^x]

 

2. (D² - 2D + 1)y = x² e^3x 

 

 3. (D² + 4D + 3) y = e^x cos 2x – cos 3x 

 

4. Find the P.I. of (D² - 4D+3) y = e^x cos 2x. 

[ Ans. P.I = -e^x/8 (sin 2x + cos 2x)

 

5. (D²+1)y = x sin hx

Hint:  

 [Ans. y = A cos x + B sin x + ½ [x sin hx cos hx]]

 

6. d²x/dt² + 2 dx/dt + 3x= t cos t

 

7. (D² + 6D +9) y = e^−2x x³ 

[Ans. y = (Ax + B) e^-3x + e^-2x (x³- 6x² + 18x-24)]

 

8. (D² + 2D - 1)y = (x + e^x)²

 

 

9. (D² + 5D + 4) y = e^-x sin 2x

 

10. (D² + D + 1) y = e^-x sin² x/2.

 

11. (D² + 2D + 5) y= e^x cos 3x

Hint: R.H.S.= e^x [ ¾ cos x + ¼ cos 3x]

 

12. (D² + 4D+8) y = 12 e^-2x sin x sin 2x

Hint: R.H.S. = 6e^-2x [cos x – cos 3x ]

 

13. dy²/dx² + 2dy/dx + y =e^-x/x² 

[ Ans. y = ( A + Bx) e^-x – e^-x log x]

 

14. d²y/dx² – 4y = x sinh x 

 

15. (D² + 4D + 4) y = e^-2x /x² 

 

16. (D² - 5D +6) y = e^x sin² x  

 

17. (D² - 3D + 2) y = xe^3x + sin 2x  [A.U A/M 2015 R-13]

 [Ans. y = C₁e^x + C₂ e^2x + e^3x/4 (2x - 3) – 1/20 ( sin 2x – 3 cos 2x)]

 

^{I. (g) R.H.S. = xn sin ax or xn cos ax Type.}

Solve the following:

1. Solve (D² - 1) y = x sin 3x + cos x.

[Ans. y =Ae^x + Be^-x – 1/10[ 3/5 cos 3x + x sin 3x + 5 cos x]]

 

2. Solve (D²-1) y = x² cos x 

[Ans. y = Ae^x + Be^-x + x sinx + 1/2(1 - x²) cos x]

 

3. Solve (D² - 4D + 4) y = 3x²e^2x sin 2x 

 

4. Solve (D² - 2D) y = e^x x² cos x

[Ans. y = A + Be^2x + e^x [(1 –x² )/2 cos x + x sin x]]

 

5. Solve (D²+ a²) y = x sin ax 

[Ans. y = A cos ax + B sin ax + 4/4a² sin ax – x²/4a cos ax ]

 

6. Solve (D² + a²) y = x cos ax 

[Ans. y = A cos ax + B sin ax + x/4a² cos ax + x²/4a sin ax]

 

7. (D² - 3D + 2) y = x cos x 

[Ans. y = C₁ex + C₂ e^2x + 1/50 [(5x - 6) cos x - (15x + 17) sin x]]

 

8. (D² + 1)2y = x² cos x 

[Ans. y = (C₁+ C₂x) cos x + (C₃ + C₄ x) sin x + 1/48 [ 4x³ sin + (9x² – x⁴) cos x ]]

 

I. (h) 1/D-a f(x) =e^ax∫ e^-ax f(x)dx Type

Solve the following:

 

1.d²y/dx² +  9y = sec 3x 

[Ans. y = C₁cos 3x + C₂ sin 3x + x/3 sin 3x + 1/9 cos 3x log cos 3x]

 

2. d²y/dx² + a²y = sec ax 

 

3. d²y/dx² + y =  cosec x 

[ Ans. C₁ cos x + C₂ sin x – x cos x.+ sin x log sin x]

 

4. d²y/dx² - 2dy/dx + 2y = e^x tan x 

[Ans. y = e^x (C₁ cosx + C₂ sin x − e^x cosx x log (sec x sin x]

 

5. d²y/dx² – 2dy/dx + 2y = e^x tan x 

[Ans. y = e^x (C₁ cos x + C₂ sin x) - e^x cos x log (sec x tan x)]

 

6. d²y/dx² + a²y =  tan ax  

[Ans. y = C₁ cos ax + C₂ sin ax – 1/a² cos ax. log (sec ax +  tan ax) ]

 

7. (D² + 4) y = tan 2x 

[Ans. y = C₁ cos 2x + C₂ sin 2x- 1/4 cos 2x log [sec 2x + tan 2x]]

 

I (i). 1. (D² + 4) y = e^y + sin 2x + cos 2x

[Ans. y = A cos 2x + B sin 2x + e^x/5 – x/4 cos 2x + x/4 sin 2x ]

 

I (j) 1. (D² − 1) y = x sin x + (1 + x²) e^x

 

 

2. (D² - 4D + 4) y = 8 (e^2x + sin 2x + x²)

 [Ans. y = (C₁+ C₂x) e^2x + 4x² e^2x + cos 2x + 2x² + 4x + 3]