1Mark Questions on First-Order Ordinary Differential Equations

1. The general solution of \(\frac{dy}{dx} = e^{x - y}\) is:

1. \(y = x + C\)
2. \(y = \ln(e^x + C)\)
3. \(y = e^x + C\)
4. \(y = -\ln(e^x + C)\)

Answer: (b) \(y = \ln(e^x + C)\)

2. An integrating factor of \(x^2 y \, dx + (y^3 - x^3) \, dy = 0\) is:

1. \(\frac{1}{x^2}\)
2. \(\frac{1}{y^2}\)
3. \(\frac{1}{x^4}\)
4. \(\frac{1}{y^4}\)

Answer: (d) \(\frac{1}{y^4}\)

3. An integrating factor of \((1 + y^2) \, dx = (\tan^{-1} y - x) \, dy\) is:

1. \(e^{\tan^{-1} y}\)
2. \(e^{y}\)
3. \(e^{x}\)
4. \(e^{\ln y}\)

Answer: (a) \(e^{\tan^{-1} y}\)

4. The differential equation of the orthogonal trajectories of the family of curves \(xy = c^2\) is:

1. \(\frac{dy}{dx} = \frac{y}{x}\)
2. \(\frac{dy}{dx} = -\frac{y}{x}\)
3. \(\frac{dy}{dx} = \frac{x}{y}\)
4. \(\frac{dy}{dx} = -\frac{x}{y}\)

Answer: (c) \(\frac{dy}{dx} = \frac{x}{y}\)

5. The differential equation satisfying the relation \(y = c e^{\sin x}\) is:

1. \(\frac{dy}{dx} = y \cos x\)
2. \(\frac{dy}{dx} = y \sin x\)
3. \(\frac{dy}{dx} = -y \cos x\)
4. \(\frac{dy}{dx} = -y \sin x\)

Answer: (a) \(\frac{dy}{dx} = y \cos x\)

6. An integrating factor of \((x - y) \, dx - dy = 0\) is:

1. \(e^x\)
2. \(e^{-x}\)
3. \(e^y\)
4. \(e^{-y}\)

Answer: (a) \(e^x\)

7. The differential equation \((\alpha x y^3 + y \cos x) \, dx + (3x^2 y^2 + \beta \sin x) \, dy = 0\) is exact if:

1. \(\alpha = 2, \beta = 1\)
2. \(\alpha = 1, \beta = 2\)
3. \(\alpha = 2, \beta = -1\)
4. \(\alpha = -2, \beta = 1\)

Answer: (a) \(\alpha = 2, \beta = 1\)

8. The general solution of \(\frac{y \, dx - x \, dy}{y^2} = 0\) is:

1. \(y = \frac{C}{x}\)
2. \(y = \frac{x}{C}\)
3. \(y = Cx^2\)
4. \(y = \frac{C}{x^2}\)

Answer: (b) \(y = \frac{x}{C}\)

9. The general solution of \(y' + y = e^{e^x}\) is:

1. \(y = e^{-x}e^{e^x} + C e^{-x}\)
2. \(y = e^{-x}e^{e^x} + C e^{x}\)
3. \(y = e^{-x}e^{e^x} + C e^{-e^x}\)
4. \(y = e^{-x}e^{e^x} + C e^{e^x}\)

Answer: (a) \(y = e^{-x}e^{e^x} + C e^{-x}\)

10. An integrating factor of the equation \(\frac{dy}{dx} - \frac{y}{x} = x\) is:

1. \(x\)
2. \(\frac{1}{x}\)
3. \(x^2\)
4. \(\frac{1}{x^2}\)

Answer: (b) \(\frac{1}{x}\)

11. The substitution that reduces \(\frac{dy}{dx} - \frac{\tan y}{1 + x} = (1 + x)e^x \sec y\) into linear form is:

1. \(u = \sin y\)
2. \(u = \cos y\)
3. \(u = \tan y\)
4. \(u = \sec y\)

Answer: (a) \(u = \sin y\)

12. The differential equation whose solution \(y = c_1 e^{-x} + c_2 e^x\) is:

1. \(\frac{d^2y}{dx^2} - y = 0\)
2. \(\frac{d^2y}{dx^2} + y = 0\)
3. \(\frac{d^2y}{dx^2} - 2y = 0\)
4. \(\frac{d^2y}{dx^2} + 2y = 0\)

Answer: (a) \(\frac{d^2y}{dx^2} - y = 0\)

13. An integrating factor of \(x \log x \frac{dy}{dx} + y = 2 \log x\) is:

1. \(\log x\)
2. \(x\)
3. \(\frac{1}{\log x}\)
4. \(\frac{1}{x}\)

Answer: (a) \(\log x\)

14. An integrating factor of \(\frac{dy}{dx} + 2xy = x e^{-x^2}\) is:

1. \(e^{x^2}\)
2. \(e^{-x^2}\)
3. \(e^{2x^2}\)
4. \(e^{-2x^2}\)

Answer: (a) \(e^{x^2}\)

15. Which of the following statement is TRUE?

1. \(f(x, y) = \sqrt{x} + \sqrt{y}\) is a homogeneous function of degree \(\sqrt{2}\)
2. \(y dx - x dy = 0\) is an exact differential equation
3. The family of con-focal and coaxial parabolas \(y^2 = 4a(x + a)\) is self-orthogonal
4. The number of arbitrary constants present in the general solution of \(y''' - 3y'' + 3y' - 1 = e^x\) is 2

Answer: (c) The family of con-focal and coaxial parabolas \(y^2 = 4a(x + a)\) is self-orthogonal

16. An integrating factor of \(\frac{dy}{dx} + y \sec^2 x = x \tan x\) is:

1. \(e^{\tan x}\)
2. \(e^{\sec x}\)
3. \(e^{\sin x}\)
4. \(e^{\cos x}\)

Answer: (a) \(e^{\tan x}\)

17. The rate at which bacteria multiply is proportional to the instantaneous number present. If the original number doubles in 3 hours, how many times the original number will be present after 9 hours?

1. 4 times
2. 6 times
3. 8 times
4. 10 times

Answer: (c) 8 times

18. The differential equation of the family of straight lines passing through the origin is:

1. \(\frac{dy}{dx} = \frac{y}{x}\)
2. \(\frac{dy}{dx} = -\frac{y}{x}\)
3. \(\frac{dy}{dx} = \frac{x}{y}\)
4. \(\frac{dy}{dx} = -\frac{x}{y}\)

Answer: (a) \(\frac{dy}{dx} = \frac{y}{x}\)

19. The general solution of \(\sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0\) is:

1. \(\tan x \tan y = C\)
2. \(\sec x \sec y = C\)
3. \(\tan x + \tan y = C\)
4. \(\sec x + \sec y = C\)

Answer: (a) \(\tan x \tan y = C\)

20. If \(e^{\int g(y) dy}\) is an integrating factor of \(\cos y \sin 2x \, dx + (\cos^2 y - \cos^2 x) \, dy = 0\), then \(g(y) =\):

1. \(\tan y - \sec y\)
2. \(\cot y+\sec y\)
3. \(\sec y - \cot y\)
4. \(\sec y + \tan y\)

Answer: (d) \(\sec y + \tan y\)

21. The value of \(\alpha\) so that \(e^{\alpha y^2}\) is an integrating factor of \(\left(xy - e^{-\frac{y^2}{2}}\right)dy + dx = 0\) is:

1. \(\frac{1}{2}\)
2. \(\frac{-1}{2}\)
3. \(\frac{1}{4}\)
4. \(\frac{1}{4}\)

Answer: (a) \(\frac{1}{2}\)

22. If \(x^\alpha\) is an integrating factor of \(\left(x + y^3\right)dx + 6xy^2 dy = 0\), then \(\alpha =\):

1. \(\frac{1}{2}\)
2. \(\frac{-1}{2}\)
3. \(\frac{1}{4}\)
4. \(\frac{-1}{4}\)

Answer: (b) \(\frac{-1}{2}\)

23. The orthogonal trajectories of the family of curves \(y = ax\) is:

1. \(y = -\frac{x}{a}\)
2. \(y = -\frac{x}{a} + C\)
3. \(x^2 + y^2 = C\)
4. \(y = Cx^2\)

Answer: (c) \(x^2 + y^2 = C\)

24. The orthogonal trajectories of the family of curves \(x^2 = y\) is:

1. \(y = -\frac{1}{2}\log x+C\)
2. \(y = \frac{1}{4} \log x+C\)
3. \(y = -\frac{1}{4} \log x+C\)
4. \(y = \frac{1}{2} \log x+C\)

Answer: (a) \(y = -\frac{1}{2}\log x+C\)

25. Which of the following statement is TRUE?

1. If \(Mdx + Ndy = 0\) is a homogeneous equation, then \((Mx - Ny)^{-1}\) is an integrating factor.
2. \(Mdx + Ndy = 0\) is an exact differential equation if \(\frac{\partial N}{\partial y} = \frac{\partial M}{\partial x}\).
3. The mathematical formulation for the law of natural growth is \(\frac{dx}{dt} = kx, \, k > 0\).
4. \(\frac{dy}{dx} - y \tan x = y^2 \sec^2 x\) is a linear equation in \(y\).

Answer: (c) The mathematical formulation for the law of natural growth is \(\frac{dx}{dt} = kx, \, k > 0\)

26. The mathematical formulation of Newton’s Law of Cooling is given by:

1. \(\frac{dT}{dt} = -k(T - T_s)\)
2. \(\frac{dT}{dt} = k(T + T_s)\)
3. \(\frac{dT}{dt} = kT(T - T_s)\)
4. \(\frac{dT}{dt} = kT(T + T_s)\)

Answer: (a) \(\frac{dT}{dt} = -k(T - T_s)\)

27. The differential equation obtained from the relation \(y = cx + c^2\) is:

1. \(y = \frac{dy}{dx}x + \left(\frac{dy}{dx}\right)^2\)
2. \(y = \frac{dy}{dx}x - \left(\frac{dy}{dx}\right)^2\)
3. \(y = \frac{dy}{dx}x + \frac{dy}{dx}\)
4. \(y = \frac{dy}{dx}x - \frac{dy}{dx}\)

Answer: (a) \(y = \frac{dy}{dx}x + \left(\frac{dy}{dx}\right)^2\)

28. The orthogonal trajectories of the family of curves \(\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1\) is:

1. \(\frac{x^2}{a^2 - \lambda} + \frac{y^2}{b^2 - \lambda} = 1\)
2. \(\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1\)
3. \(\frac{x^2}{a^2 - \lambda} - \frac{y^2}{b^2 - \lambda} = 1\)
4. \(\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 - \lambda} = 1\)

Answer: (b) \(\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1\)

29. The orthogonal trajectories of the family of curves \(y^2 = x\) is:

1. \(\frac{1}{2}\log y = -x + C\)
2. \(\frac{1}{2}\log y = x + C\)
3. \(\frac{1}{2}\log y = 2x + C\)
4. \(\frac{1}{2}\log y = -2x + C\)

Answer: (a) \(\frac{1}{2}\log y = -x + C\)

30. An integrating factor of \(\left(y + 1 + x^2\right)dx + \left(x^2 \sin y - x\right)dy = 0\) is:

1. \(\frac{1}{x}\)
2. \(\frac{1}{x^2}\)
3. \(\frac{-1}{x}\)
4. \(\frac{-1}{x^2}\)

Answer: (b) \(\frac{1}{x^2}\)

31. For what values of \(p\) is the equation \(\left(x^2 - p^2 y\right)dx + \left(y^2 - 3x\right)dy = 0\) exact?

1. \(p = 3\)
2. \(p = \frac{1}{\sqrt{3}}\)
3. \(p = \sqrt{3}\)
4. \(p = -\sqrt{3}\)

Answer: (c) \(p = \sqrt{3}\)

32. An integrating factor of \(y\left(2x^2 - xy + 1\right)dx + \left(x - y\right)dy = 0\) is:

1. \(e^x\)
2. \(e^{-x}\)
3. \(e^{x^2}\)
4. \(e^{-x^2}\)

Answer: (c) \(e^{x^2}\)

33. The order of a differential equation obtained from \(y = a e^x + b e^{2x} + c e^{3x}\), where \(a, b, c\) are arbitrary constants, is:

1. 1
2. 2
3. 3
4. 4

Answer: (c) 3

34. Which of the following is not an integrating factor of \(y dx - x dy = 0\)?

1. \(\frac{1}{x^3}\)
2. \(\frac{1}{y^2}\)
3. \(\frac{1}{x^2y}\)
4. \(\frac{-1}{x^2 y^2}\)

Answer: (b) \(\frac{1}{y^2}\)

35. The general solution of \(\frac{dy}{dx} + 2xy = e^{-x^2}\) is:

1. \(y = e^{-x^2}(x + C)\)
2. \(y = e^{x^2}(x + C)\)
3. \(y = e^{-x^2}(x^2 + C)\)
4. \(y = e^{x^2}(x^2 + C)\)

Answer: (a) \(y = e^{-x^2}(x + C)\)

36. An integrating factor of \(x\left(xy \sin xy + \cos xy\right)dx + x\left(xy \sin xy - \cos xy\right)dy = 0\) is:

1. \(\frac{-1}{xy\cos xy}\)
2. \(\frac{-1}{2xy\cos xy}\)
3. \(\frac{1}{2xy\cos xy}\)
4. \(\frac{1}{xy\cos xy}\)

Answer: (c) \(\frac{1}{2xy\cos xy}\)

37. The population of a city gets doubled in 50 years. In how many years will it treble under the assumption that the rate of increase is proportional to the number of inhabitants?

1. 79.24 years
2. 80.24 years
3. 85.24 years
4. 78.24 years

Answer: (a) 79.24 years

38. The maximum number of linearly independent solutions of the differential equation \(\frac{d^4y}{dx^4} = 0\) is:

1. 1
2. 2
3. 3
4. 4

Answer: (d) 4

39. The general solution of \(\log\left(\frac{dy}{dx}\right) = 2x + 3y\) is:

1. \(3e^{2x}+2e^{y} = C\)
2. \(3e^{2x}+2e^{3y} = C\)
3. \(3e^{x}+2e^{3y} = C\)
4. \(3e^{2x}+2e^{y} = C\)

Answer: (b) \(3e^{2x}+2e^{3y} = C\)

40. If \(\phi(x)\) is a function such that \(\frac{d\phi}{dx} = 2\phi\), then \(\phi(x) =\):

1. \(Ce^{2x}\)
2. \(Ce^{-2x}\)
3. \(Cx^2\)
4. \(Cx^{-2}\)

Answer: (a) \(Ce^{2x}\)

41. The necessary and sufficient condition for the differential equation \(\phi(x, y)dx + \psi(x, y)dy = 0\) to be exact is:

1. \(\frac{\partial \phi}{\partial y} = \frac{\partial \psi}{\partial x}\)
2. \(\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}\)
3. \(\frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}\)
4. \(\frac{\partial \phi}{\partial x} = -\frac{\partial \psi}{\partial y}\)

Answer: (a) \(\frac{\partial \phi}{\partial y} = \frac{\partial \psi}{\partial x}\)

42. The value of \(\lambda\) such that the differential equation \(\left(xy^2 - 2\lambda y\right)dx + \left(x^2y - 6x\right)dy = 0\) is exact:

1. \(\lambda = 2\)
2. \(\lambda = 6\)
3. \(\lambda = 3\)
4. \(\lambda = 4\)

Answer: (c) \(\lambda = 3\)

43. The orthogonal trajectories of the family of parabolas \(y^2 = 4a(x + a)\) is:

1. \(y^2 = -4a(x + a)\)
2. \(y^2 = 4a(x - a)\)
3. \(y^2 = -4a(x - a)\)
4. \(y^2 = 4a(x + a)\)

Answer: (d) \(y^2 = 4a(x + a)\)

44. A bacterial culture growing exponentially increased from 100 to 250 grams in the period from 6 a.m to 9 a.m. How many grams will be present at noon?

1. 500 grams
2. 625 grams
3. 750 grams
4. 875 grams

Answer: (b) 625 grams

45. If the temperature of the air is 290K and the object cools from 370K to 330K in 10 minutes, then the temperature of the object after 20 minutes will be:

1. 300K
2. 310K
3. 320K
4. 330K

Answer: (b) 310K

46. If \(\frac{dy}{dx} - x^2 y = y^2 e^{-x^3/3}\) can be reduced to the linear form \(\frac{dt}{dx} + x^2 t = -e^{-x^3/3}\), then \(t =\):

1. \(\frac{1}{y}\)
2. \(\frac{1}{y^2}\)
3. \(\frac{1}{y^3}\)
4. \(\frac{1}{y^4}\)

Answer: (a) \(\frac{1}{y}\)

47. The solution of \(y dx - x dy = 3x^2 e^{x^3} y^2 dx\) is:

1. \(\frac{x}{y} + e^{x^3}+C\)
2. \(\frac{y}{x} + e^{-x^3}+C\)
3. \(\frac{x}{y} + e^{-x^3}+C\)
4. \(\frac{y}{x} + e^{x^3}+C\)

Answer: (a) \(\frac{x}{y} + e^{x^3}+C\)

48. Which of the following equations is not exact?

1. \(\left(2x + e^x \log y\right)dx + \left(\frac{e^x}{y} + 1\right)dy = 0\)
2. \(\left(y + \log x\right)dx - x dy = 0\)
3. \(\left(ax + hy + g\right)dx + \left(hx + by + f\right)dy = 0\)
4. \(\left(e^y + 1\right)\cos x dx + e^y \sin x dy = 0\)

Answer: (b) \(\left(y + \log x\right)dx - x dy = 0\)

49. If \(\frac{1}{N}\left(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}\right) = k\), where \(k\) is a constant, then the integrating factor of \(M dx + N dy = 0\) is:

1. \(e^{kx}\)
2. \(e^{ky}\)
3. \(e^{kxy}\)
4. \(e^{k(x + y)}\)

Answer: (a) \(e^{kx}\)