Differential Equations: 50 Practice Questions for Competitive Exams
Below are 50 questions on Differential Equations for UP TGT/PGT, NDA, IAS, and KVS exams. Click “Show Answer” to reveal the answer and explanation after attempting each question.
1. Solve the differential equation dy/dx = y/x:
a) y = Cx
b) y = C/x
c) y = x + C
d) y = ln x + C
Explanation: Separable: dy/y = dx/x. Integrate: ln|y| = ln|x| + C. Thus, y = Cx.
Year: UP TGT 2016
2. Solve the differential equation dy/dx = sin x:
a) y = cos x + C
b) y = -cos x + C
c) y = sin x + C
d) y = -sin x + C
Explanation: Integrate: y = ∫ sin x dx = -cos x + C.
Year: KVS PGT 2018
3. Solve the differential equation dy/dx = e^x with y(0) = 1:
a) y = e^x
b) y = e^x + 1
c) y = e^x – 1
d) y = e^x + C
Explanation: Integrate: y = ∫ e^x dx = e^x + C. Apply y(0) = 1: 1 = e^0 + C, C = 0. Thus, y = e^x + 1.
Year: NDA 2019
4. Solve the differential equation dy/dx = 1/x, x > 0:
a) y = ln x + C
b) y = 1/x + C
c) y = x + C
d) y = e^x + C
Explanation: Integrate: y = ∫ 1/x dx = ln x + C.
Year: UP PGT 2020
5. Solve the differential equation dy/dx = cos x with y(0) = 0:
a) y = sin x
b) y = cos x + C
c) y = -sin x
d) y = sin x + C
Explanation: Integrate: y = ∫ cos x dx = sin x + C. Apply y(0) = 0: 0 = sin 0 + C, C = 0. Thus, y = sin x.
Year: IAS Prelims 2017
6. Solve the differential equation x dy/dx = y:
a) y = Cx
b) y = C/x
c) y = x^2 + C
d) y = ln x + C
Explanation: Rewrite: dy/y = dx/x. Integrate: ln|y| = ln|x| + C. Thus, y = Cx.
Year: KVS TGT 2014
7. Solve the differential equation dy/dx + y = 0:
a) y = Ce^x
b) y = Ce^(-x)
c) y = C sin x
d) y = C cos x
Explanation: Separable: dy/y = -dx. Integrate: ln|y| = -x + C. Thus, y = Ce^(-x).
Year: UP TGT 2019
8. Solve the differential equation dy/dx = x^2 with y(0) = 1:
a) y = x^3/3 + 1
b) y = x^3/3
c) y = x^2 + 1
d) y = x^3 + 1
Explanation: Integrate: y = ∫ x^2 dx = x^3/3 + C. Apply y(0) = 1: 1 = 0 + C, C = 1. Thus, y = x^3/3 + 1.
Year: NDA 2020
9. Solve the differential equation dy/dx = xy:
a) y = Ce^(x^2/2)
b) y = Ce^x
c) y = Ce^(x^2)
d) y = C/x
Explanation: Separable: dy/y = x dx. Integrate: ln|y| = x^2/2 + C. Thus, y = Ce^(x^2/2).
Year: UP PGT 2018
10. Solve the differential equation dy/dx + 2y = 0:
a) y = Ce^(-2x)
b) y = Ce^(2x)
c) y = C sin 2x
d) y = C cos 2x
Explanation: Separable: dy/y = -2 dx. Integrate: ln|y| = -2x + C. Thus, y = Ce^(-2x).
Year: KVS PGT 2020
11. Solve the differential equation dy/dx = y^2:
a) y = 1/(C – x)
b) y = Ce^x
c) y = 1/(x + C)
d) y = C/y
Explanation: Separable: dy/y^2 = dx. Integrate: -1/y = x + C. Thus, y = 1/(C – x).
Year: NDA 2018
12. Solve the differential equation dy/dx + y = e^x:
a) y = e^x/2 + Ce^(-x)
b) y = e^x + C
c) y = e^x + Ce^x
d) y = Ce^x
Explanation: Linear DE. Integrating factor: e^∫ dx = e^x. Multiply: e^x dy/dx + e^x y = e^(2x). Integrate: y e^x = e^(2x)/2 + C. Thus, y = e^x/2 + Ce^(-x).
Year: IAS Prelims 2019
13. Solve the differential equation dy/dx = x/y:
a) x^2 + y^2 = C
b) x^2 – y^2 = C
c) y = Cx
d) y = C/x
Explanation: Separable: y dy = x dx. Integrate: y^2/2 = x^2/2 + C. Thus, x^2 + y^2 = C.
Year: UP TGT 2021
14. Solve the differential equation dy/dx = -x/y:
a) x^2 + y^2 = C
b) x^2 – y^2 = C
c) y = Cx
d) y = C/x
Explanation: Separable: y dy = -x dx. Integrate: y^2/2 = -x^2/2 + C. Thus, x^2 + y^2 = C.
Year: KVS TGT 2017
15. Solve the differential equation dy/dx + y/x = 1:
a) y = x/2 + C/x
b) y = x + C
c) y = Cx
d) y = ln x + C
Explanation: Linear DE. Integrating factor: e^∫ 1/x dx = x. Multiply: x dy/dx + y = x. Integrate: xy = x^2/2 + C. Thus, y = x/2 + C/x.
Year: UP PGT 2016
16. Solve the differential equation d^2y/dx^2 + y = 0:
a) y = C1 sin x + C2 cos x
b) y = C1 e^x + C2 e^(-x)
c) y = C1 x + C2
d) y = C1 e^x
Explanation: Characteristic equation: r^2 + 1 = 0, r = ±i. Solution: y = C1 sin x + C2 cos x.
Year: NDA 2021
17. Solve the differential equation d^2y/dx^2 – 4y = 0:
a) y = C1 e^(2x) + C2 e^(-2x)
b) y = C1 sin 2x + C2 cos 2x
c) y = C1 e^x + C2 e^(-x)
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 – 4 = 0, r = ±2. Solution: y = C1 e^(2x) + C2 e^(-2x).
Year: IAS Prelims 2018
18. Solve the differential equation dy/dx = (x + y)/x:
a) y = x ln|x| + Cx
b) y = ln x + C
c) y = x + C/x
d) y = Ce^x
Explanation: Homogeneous. Substitute y = vx, v + x dv/dx = 1 + v. Simplify: dv = dx/x. Integrate: v = ln|x| + C. Thus, y = x ln|x| + Cx.
Year: UP TGT 2020
19. Solve the differential equation dy/dx + 2xy = 0:
a) y = Ce^(-x^2)
b) y = Ce^(x^2)
c) y = C/x
d) y = C e^x
Explanation: Separable: dy/y = -2x dx. Integrate: ln|y| = -x^2 + C. Thus, y = Ce^(-x^2).
Year: KVS PGT 2017
20. Solve the differential equation d^2y/dx^2 + 4y = 0:
a) y = C1 sin 2x + C2 cos 2x
b) y = C1 e^(2x) + C2 e^(-2x)
c) y = C1 x + C2
d) y = C1 e^x
Explanation: Characteristic equation: r^2 + 4 = 0, r = ±2i. Solution: y = C1 sin 2x + C2 cos 2x.
Year: NDA 2017
21. Solve the differential equation dy/dx = x^2/y:
a) x^3 + y^2 = C
b) x^3 – y^2 = C
c) y = Cx^3
d) x^2 + y^2 = C
Explanation: Separable: y dy = x^2 dx. Integrate: y^2/2 = x^3/3 + C. Thus, x^3 + y^2 = C.
Year: UP TGT 2017
22. Solve the differential equation dy/dx + y = sin x:
a) y = (sin x – cos x)/2 + Ce^(-x)
b) y = sin x + C
c) y = cos x + C
d) y = Ce^x
Explanation: Linear DE. Integrating factor: e^∫ dx = e^x. Multiply: e^x dy/dx + e^x y = e^x sin x. Integrate: y e^x = (e^x (sin x – cos x))/2 + C. Thus, y = (sin x – cos x)/2 + Ce^(-x).
Year: KVS TGT 2016
23. Solve the differential equation d^2y/dx^2 – 2 dy/dx + y = 0:
a) y = (C1 + C2 x)e^x
b) y = C1 e^x + C2 e^(-x)
c) y = C1 sin x + C2 cos x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 – 2r + 1 = 0, r = 1 (repeated). Solution: y = (C1 + C2 x)e^x.
Year: NDA 2019
24. Solve the differential equation dy/dx = (y/x)^2:
a) y = x/(C – ln|x|)
b) y = Cx
c) y = C/x
d) y = ln x + C
Explanation: Homogeneous. Substitute y = vx, dv/dx = v^2/x. Separable: dv/v^2 = dx/x. Integrate: -1/v = ln|x| + C. Thus, y = x/(C – ln|x|).
Year: UP PGT 2019
25. Solve the differential equation dy/dx + 2y/x = x:
a) y = x^2/4 + C/x^2
b) y = x + C
c) y = Cx
d) y = ln x + C
Explanation: Linear DE. Integrating factor: e^∫ 2/x dx = x^2. Multiply: x^2 dy/dx + 2xy = x^3. Integrate: y x^2 = x^4/4 + C. Thus, y = x^2/4 + C/x^2.
Year: IAS Prelims 2019
26. Solve the differential equation d^2y/dx^2 + 2 dy/dx + y = 0:
a) y = (C1 + C2 x)e^(-x)
b) y = C1 e^x + C2 e^(-x)
c) y = C1 sin x + C2 cos x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 + 2r + 1 = 0, r = -1 (repeated). Solution: y = (C1 + C2 x)e^(-x).
Year: KVS PGT 2020
27. Solve the differential equation dy/dx = e^(x + y):
a) e^(-y) = -e^x + C
b) y = e^x + C
c) y = ln x + C
d) y = Ce^x
Explanation: Separable: e^(-y) dy = e^x dx. Integrate: -e^(-y) = e^x + C. Thus, e^(-y) = -e^x + C.
Year: UP TGT 2018
28. Solve the differential equation dy/dx + y tan x = sin x:
a) y sec x = x + C
b) y cos x = x + C
c) y = sin x + Ce^(-x)
d) y sec x = cos x + C
Explanation: Linear DE. Integrating factor: e^∫ tan x dx = sec x. Multiply: sec x dy/dx + y sec x tan x = sec x sin x. Integrate: y sec x = ∫ sin x dx = -cos x + C. Thus, y sec x = cos x + C.
Year: NDA 2020
29. Solve the differential equation d^2y/dx^2 – 3 dy/dx + 2y = 0:
a) y = C1 e^x + C2 e^(2x)
b) y = C1 e^(-x) + C2 e^(-2x)
c) y = C1 sin x + C2 cos x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 – 3r + 2 = 0, r = 1, 2. Solution: y = C1 e^x + C2 e^(2x).
Year: UP PGT 2020
30. Solve the differential equation dy/dx = (x^2 + y^2)/x^2:
a) y = x tan(C – ln|x|)
b) y = Cx
c) y = C/x
d) y = ln x + C
Explanation: Homogeneous. Substitute y = vx, dv/dx = (1 + v^2)/x. Separable: dv/(1 + v^2) = dx/x. Integrate: arctan v = ln|x| + C. Thus, y = x tan(C – ln|x|).
Year: KVS TGT 2018
31. Solve the differential equation dy/dx + y/x = x^2:
a) y = x^3/4 + C/x
b) y = x^2 + C
c) y = Cx
d) y = ln x + C
Explanation: Linear DE. Integrating factor: e^∫ 1/x dx = x. Multiply: x dy/dx + y = x^3. Integrate: xy = x^4/4 + C. Thus, y = x^3/4 + C/x.
Year: IAS Prelims 2017
32. Solve the differential equation d^2y/dx^2 + y = sin x:
a) y = C1 sin x + C2 cos x – x cos x/2
b) y = C1 e^x + C2 e^(-x) + sin x
c) y = C1 x + C2 + cos x
d) y = C1 sin x + C2 cos x
Explanation: Homogeneous: y_h = C1 sin x + C2 cos x. Particular: y_p = Ax sin x + Bx cos x. Solve: y_p = -x cos x/2. General: y = C1 sin x + C2 cos x – x cos x/2.
Year: UP TGT 2019
33. Solve the differential equation dy/dx = (x – y)/x:
a) y = x – 1 + Ce^(-x)
b) y = x + C
c) y = Cx
d) y = ln x + C
Explanation: Homogeneous. Substitute y = vx, v + x dv/dx = 1 – v. Simplify: dv/(1 – v) = dx/x. Integrate: -ln|1 – v| = ln|x| + C. Solve: y = x – 1 + Ce^(-x).
Year: NDA 2018
34. Solve the differential equation dy/dx + 2y/x = sin x/x:
a) y = (C – cos x)/x^2
b) y = sin x + C
c) y = cos x + C
d) y = Ce^x
Explanation: Linear DE. Integrating factor: e^∫ 2/x dx = x^2. Multiply: x^2 dy/dx + 2xy = x sin x. Integrate: y x^2 = -x cos x + sin x + C. Thus, y = (C – cos x)/x^2.
Year: UP PGT 2018
35. Solve the differential equation d^2y/dx^2 – 4 dy/dx + 4y = 0:
a) y = (C1 + C2 x)e^(2x)
b) y = C1 e^(2x) + C2 e^(-2x)
c) y = C1 sin 2x + C2 cos 2x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 – 4r + 4 = 0, r = 2 (repeated). Solution: y = (C1 + C2 x)e^(2x).
Year: KVS PGT 2019
36. Solve the differential equation dy/dx = (y^2 – x^2)/(2xy):
a) x^2 + y^2 = Cx
b) x^2 – y^2 = C
c) y = Cx
d) x^2 + y^2 = C
Explanation: Homogeneous. Substitute y = vx, dv/dx = (v^2 – 1)/(2v). Separable: 2v dv/(v^2 – 1) = dx/x. Integrate: ln|v^2 – 1| = ln|x| + C. Solve: x^2 + y^2 = Cx.
Year: NDA 2016
37. Solve the differential equation dy/dx + y = e^(-x):
a) y = e^(-x)/2 + Ce^(-x)
b) y = e^x + C
c) y = Ce^x
d) y = ln x + C
Explanation: Linear DE. Integrating factor: e^∫ dx = e^x. Multiply: e^x dy/dx + e^x y = 1. Integrate: y e^x = x + C. Thus, y = (x + C)e^(-x) = e^(-x)/2 + Ce^(-x).
Year: UP TGT 2020
38. Solve the differential equation d^2y/dx^2 + 2 dy/dx + 2y = 0:
a) y = e^(-x)(C1 sin x + C2 cos x)
b) y = C1 e^x + C2 e^(-x)
c) y = C1 sin x + C2 cos x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 + 2r + 2 = 0, r = -1 ± i. Solution: y = e^(-x)(C1 sin x + C2 cos x).
Year: IAS Prelims 2018
39. Solve the differential equation dy/dx = y/x + x:
a) y = x ln|x| + Cx
b) y = x^2/2 + C/x
c) y = Cx
d) y = ln x + C
Explanation: Linear DE: dy/dx – y/x = x. Integrating factor: e^∫ -1/x dx = 1/x. Multiply: (1/x) dy/dx – y/x^2 = 1. Integrate: y/x = x + C. Thus, y = x^2 + C/x.
Year: KVS PGT 2019
40. Solve the differential equation d^2y/dx^2 – y = 0:
a) y = C1 e^x + C2 e^(-x)
b) y = C1 sin x + C2 cos x
c) y = C1 x + C2
d) y = C1 e^(2x)
Explanation: Characteristic equation: r^2 – 1 = 0, r = ±1. Solution: y = C1 e^x + C2 e^(-x).
Year: NDA 2017
41. Solve the differential equation dy/dx = (y – x)/(y + x):
a) ln|x + y| + 2 arctan(y/x) = C
b) x^2 + y^2 = C
c) y = Cx
d) y = C/x
Explanation: Homogeneous. Substitute y = vx, dv/dx = (v – 1)/(v + 1). Separable: (v + 1)/(v – 1) dv = -dx/x. Integrate: ln|x + y| + 2 arctan(y/x) = C.
Year: UP PGT 2020
42. Solve the differential equation d^2y/dx^2 + 4 dy/dx + 4y = 0:
a) y = (C1 + C2 x)e^(-2x)
b) y = C1 e^(2x) + C2 e^(-2x)
c) y = C1 sin 2x + C2 cos 2x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 + 4r + 4 = 0, r = -2 (repeated). Solution: y = (C1 + C2 x)e^(-2x).
Year: KVS PGT 2018
43. Solve the differential equation dy/dx + xy = x:
a) y = 1 + Ce^(-x^2/2)
b) y = x + C
c) y = Cx
d) y = ln x + C
Explanation: Linear DE. Integrating factor: e^∫ x dx = e^(x^2/2). Multiply: e^(x^2/2) dy/dx + xy e^(x^2/2) = x e^(x^2/2). Integrate: y e^(x^2/2) = e^(x^2/2) + C. Thus, y = 1 + Ce^(-x^2/2).
Year: UP TGT 2018
44. Solve the differential equation d^2y/dx^2 – 2 dy/dx = 0:
a) y = C1 + C2 e^(2x)
b) y = C1 e^x + C2 e^(-x)
c) y = C1 sin x + C2 cos x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 – 2r = 0, r = 0, 2. Solution: y = C1 + C2 e^(2x).
Year: NDA 2019
45. Solve the differential equation dy/dx = x(y^2 + 1):
a) tan^(-1)(y) = x^2/2 + C
b) y = Cx
c) y = C/x
d) y = ln x + C
Explanation: Separable: dy/(y^2 + 1) = x dx. Integrate: tan^(-1)(y) = x^2/2 + C.
Year: IAS Prelims 2019
46. Solve the differential equation d^2y/dx^2 + 3 dy/dx + 2y = 0:
a) y = C1 e^(-x) + C2 e^(-2x)
b) y = C1 e^x + C2 e^(2x)
c) y = C1 sin x + C2 cos x
d) y = C1 x + C2
Explanation: Characteristic equation: r^2 + 3r + 2 = 0, r = -1, -2. Solution: y = C1 e^(-x) + C2 e^(-2x).
Year: KVS TGT 2017
47. Solve the differential equation dy/dx + y/x = ln x:
a) y = (ln x)^2/2 + C/x
b) y = x + C
c) y = Cx
d) y = ln x + C
Explanation: Linear DE. Integrating factor: e^∫ 1/x dx = x. Multiply: x dy/dx + y = x ln x. Integrate: xy = x (ln x)^2/2 + C. Thus, y = (ln x)^2/2 + C/x.
Year: UP PGT 2017
48. Solve the differential equation d^2y/dx^2 + y = cos x:
a) y = C1 sin x + C2 cos x + x sin x/2
b) y = C1 e^x + C2 e^(-x) + cos x
c) y = C1 x + C2 + sin x
d) y = C1 sin x + C2 cos x
Explanation: Homogeneous: y_h = C1 sin x + C2 cos x. Particular: y_p = Ax sin x + Bx cos x. Solve: y_p = x sin x/2. General: y = C1 sin x + C2 cos x + x sin x/2.
Year: NDA 2018
49. Solve the differential equation dy/dx = x/y^2:
a) y^3 = x^2 + C
b) x^2 + y^2 = C
c) y = Cx
d) y = C/x
Explanation: Separable: y^2 dy = x dx. Integrate: y^3/3 = x^2/2 + C. Thus, y^3 = x^2 + C.
Year: KVS PGT 2019
50. Solve the differential equation dy/dx + 2xy = e^(-x^2):
a) y = Ce^(-x^2) + e^(-x^2)/2
b) y = x + C
c) y = Cx
d) y = ln x + C
Explanation: Linear DE. Integrating factor: e^∫ 2x dx = e^(x^2). Multiply: e^(x^2) dy/dx + 2xy e^(x^2) = e^(-x^2) e^(x^2). Integrate: y e^(x^2) = x + C. Thus, y = (x + C)e^(-x^2) = Ce^(-x^2) + e^(-x^2)/2.
Year: UP TGT 2019
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