31. The general solution of $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{y}} = 0$$ is
32. The solution for the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {{\text{x}}^2}{\text{y}}$$ with the condition that y = 1 at x = 0 is
33. The solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\text{ky}},\,{\text{y}}\left( 0 \right) = {\text{c}}$$ is
34. The solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + {{\text{y}}^2} = 0$$ is
35. If y = 2x3 - 3x2 + 3x - 10, the value of ∆3y will be (where, ∆ is forward differences operator)
36. Match List-I with List-II and select the correct answer:
List-I
List-II
A. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{y}}}{{\text{x}}}$$
1. Circles
B. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{y}}}{{\text{x}}}$$
2. Straight lines
C. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{x}}}{{\text{y}}}$$
3. Hyperbolas
D. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{x}}}{{\text{y}}}$$
| List-I | List-II |
| A. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{y}}}{{\text{x}}}$$ | 1. Circles |
| B. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{y}}}{{\text{x}}}$$ | 2. Straight lines |
| C. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{x}}}{{\text{y}}}$$ | 3. Hyperbolas |
| D. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{x}}}{{\text{y}}}$$ |
37. If a and b are constants, the most general solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 2\frac{{{\text{dx}}}}{{{\text{dt}}}} + {\text{x}} = 0$$ is
38. The solution of the differential equation, for t > 0, y''(t) + 2y'(t) + y(t) = 0 with initial conditions
y(0) = 0 and y'(0) = 1, is (u(t) denotes the unit step function),
39. A differential equation is given as
$${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 2{\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{y}} = 4$$
The solution of differential equation in terms of arbitrary constant C1 and C2 is
$${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 2{\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{y}} = 4$$
The solution of differential equation in terms of arbitrary constant C1 and C2 is
40. The solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 0$$ with boundary conditions
$${\text{i}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 0;\,{\text{ii}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 1{\text{ is}}$$
$${\text{i}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 0;\,{\text{ii}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 1{\text{ is}}$$