1.Find the order and degree of the differential equation $\frac{d^2y}{dx^2} = \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2}$.
2.Determine the degree of the DE: $\sin^{-1}\left(\frac{d^2y}{dx^2}\right) = x + y$.
3.Find the order of the differential equation whose general solution is $y = (c_1+c_2) \cos(x+c_3) + c_4 e^{x+c_5}$.
4.Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
5.Find the general solution of the differential equation $\frac{dy}{dx} = \frac{x^2+y^2+1}{2xy}$. (Hint: Reducible to homogeneous or exact).
6.Solve: $\frac{dy}{dx} = \sin(x+y) + \cos(x+y)$.
7.Solve the homogeneous differential equation $x \frac{dy}{dx} = y + \sqrt{x^2+y^2}$.
8.Solve: $(x^2+y^2) dx + 3xy dy = 0$.
9.Find the Integrating Factor of the differential equation $(x+2y^3) \frac{dy}{dx} = y$.
10.Solve: $\frac{dy}{dx} + y \tan x = y^3 \sec x$. (Bernoulli's Equation).
11.Find the orthogonal trajectories of the family of rectangular hyperbolas $xy = c^2$.
12.Solve: $x dy - y dx = \sqrt{x^2+y^2} dx$.
13.Find the particular solution of $\log\left(\frac{dy}{dx}\right) = 3x+4y$ given that $y=0$ when $x=0$.
14.Solve: $\frac{dy}{dx} = \frac{1}{x \cos y + \sin 2y}$.
15.If $y_1$ and $y_2$ are two solutions of the linear differential equation $\frac{dy}{dx} + Py = Q$, find the general solution in terms of $y_1$ and $y_2$.
16.A normal is drawn to a curve at point $P(x,y)$. It meets the x-axis at $G$. If $OG^2 = OX^2 + OY^2$, form the differential equation.
17.Solve: $\frac{dy}{dx} + \frac{y}{x} \log y = \frac{y}{x^2} (\log y)^2$.
18.Determine the values of $m$ for which $y = e^{mx}$ is a solution of $y'' - 5y' + 6y = 0$.
19.Solve: $x^2 \frac{dy}{dx} - xy = 1 + \cos(y/x)$.
20.Find the general solution of $(x^2+y^2+2x) dx + 2y dy = 0$.
21.Solve the DE: $y(2xy + e^x) dx - e^x dy = 0$.
22.Find the orthogonal trajectories of the family of circles touching the x-axis at the origin.
23.Solve: $\frac{dy}{dx} = \frac{y^2-x^2}{2xy}$.
24.Find the curve such that the area of the triangle formed by the tangent, the x-axis, and the ordinate at any point is constant.
25.Solve: $\frac{dy}{dx} + \frac{x}{1-x^2} y = x\sqrt{y}$.
26.Form the DE of all circles touching the line $y=x$ at the origin.
27.Solve: $(1+x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0$.
28.Find the general solution of $y^2 dx + (x^2 - xy + y^2) dy = 0$.
29.Solve: $\frac{dy}{dx} = \frac{x+y+1}{2x+2y+3}$.
30.Solve the exact differential equation: $(2xy + y - \tan y) dx + (x^2 - x \tan^2 y + \sec^2 y) dy = 0$.
31.Find the curve passing through $(1, \pi/4)$ such that the slope of the tangent at any point $(x,y)$ is $\frac{y}{x} - \cos^2(y/x)$.
32.Solve: $\frac{dy}{dx} = \frac{y}{x + y^{1/3}}$.
33.Solve: $x^3 \frac{dy}{dx} - x^2 y = -y^4 \cos x$.
34.The population of a village increases at a rate proportional to the product of its current population and the difference between its maximum capacity ($L$) and current population. Form the DE.
35.Solve the DE: $\frac{dy}{dx} + \frac{y}{x} \log y = \frac{y}{x^2} (\log y)^2$.
36.Find the general solution of $(x \cos y - y \sin y) dy + (x \sin y + y \cos y) dx = 0$.
37.Solve: $\frac{dy}{dx} = \frac{2y}{x} + x^3$.
38.Find the equation of the curve where the subnormal is constant.
39.Solve: $y(1+xy) dx + x(1-xy) dy = 0$.
40.Solve: $\frac{dy}{dx} + \frac{y}{1+x^2} = \frac{e^{\tan^{-1}x}}{1+x^2}$.
41.Find the particular solution of $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$ given $y=0$ when $x=0$.
42.Solve: $\sin y \frac{dy}{dx} = \cos x (2 - \sin y \sec x)$.
43.Solve the homogeneous DE: $(x^3 - 3xy^2) dx = (y^3 - 3x^2y) dy$.
44.Find the orthogonal trajectories of $y^2 = 4ax + 4a^2$.
45.Solve: $\frac{dy}{dx} + y = \frac{1+x}{x} e^{-x}$.
46.Solve: $x dy = (y + x e^{y/x}) dx$.
47.Form the DE of the family of parabolas having their axes of symmetry coinciding with the x-axis.
48.Solve: $\frac{dy}{dx} + x \sin 2y = x^3 \cos^2 y$.
49.Find the particular solution of $(1+x^2) dy = (e^x - y) dx$ if $y(0) = 0$.
50.Solve the DE: $x \frac{dy}{dx} + y = y^2 \log x$.
51.Solve: $\frac{dy}{dx} = \frac{y}{x} + x \tan(y/x)$.
52.Determine the curve passing through $(1,0)$ if the slope of the tangent at $(x,y)$ is $\frac{y-1}{x^2+x}$.
53.Solve: $(x^2+y^2) dy = xy dx$.
54.Find the general solution of $y' + y \sec^2 x = \tan x \sec^2 x$.
55.Solve: $\frac{dy}{dx} + \frac{x}{1+x^2} y = \frac{1}{x(1+x^2)}$.
56.Solve: $e^y dx + (x e^y + 2y) dy = 0$.
57.Determine the orthogonal trajectories of the family of cardioids $r = a(1 + \cos \theta)$.
58.Solve: $\frac{dy}{dx} = \frac{y}{x} + \frac{\phi(y/x)}{\phi'(y/x)}$.
59.Solve: $\frac{dy}{dx} + \frac{y}{x} = x^n$.
60.Find the particular solution of $(x-y)(dx+dy) = dx - dy$ if $y = -1$ when $x = 0$.
