1.Find the order and degree of the DE: $y = x \frac{dy}{dx} + \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$.
2.Determine the degree of: $e^{\frac{d^3y}{dx^3}} - x \frac{d^2y}{dx^2} + y = 0$.
3.Find the order of the differential equation of the family of circles having center on the y-axis and passing through the origin.
4.Find the degree of the DE: $\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = \rho \frac{d^2y}{dx^2}$.
5.Form the DE of the family of parabolas having vertex at the origin and axis along the positive x-axis.
6.If $y = a \cos(x+b)$, form the corresponding differential equation by eliminating constants $a$ and $b$.
7.Verify that $y = c_1 e^{ax} \cos bx + c_2 e^{ax} \sin bx$ is the general solution of $y'' - 2ay' + (a^2+b^2)y = 0$.
8.Form the differential equation representing the family of ellipses having foci on the x-axis and center at the origin.
9.State the order and degree of: $\log_e\left(\frac{d^2y}{dx^2}\right) = x$.
10.Find the order of the DE whose general solution is $y = (c_1+c_2) e^x + c_3 e^{x+c_4}$.
11.Solve: $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$.
12.Find the general solution of: $\frac{dy}{dx} = \sin^{-1} x$.
13.Solve: $(1+e^{2x}) dy + (1+y^2) e^x dx = 0$.
14.Find the particular solution of $\frac{dy}{dx} = y \tan x$ given that $y=1$ when $x=0$.
15.Solve: $\frac{dy}{dx} = (4x+y+1)^2$.
16.Solve the DE: $\frac{dy}{dx} + \sqrt{\frac{1-y^2}{1-x^2}} = 0$.
17.Find the equation of the curve passing through $(1,1)$ whose differential equation is $x dy = (2x^2+1) dx$ ($x \neq 0$).
18.Solve: $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$.
19.Solve: $\cos x \cos y \frac{dy}{dx} = -\sin x \sin y$.
20.Find the general solution of $\frac{dy}{dx} = \frac{x(2 \log x + 1)}{\sin y + y \cos y}$.
21.Solve: $\frac{dy}{dx} = \cos(x+y) + \sin(x+y)$.
22.Solve: $x^2(y+1) dx + y^2(x-1) dy = 0$.
23.Show that $(x^2-y^2) dx + 2xy dy = 0$ is homogeneous and solve it.
24.Solve: $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$.
25.Solve: $\frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right)$.
26.Find the general solution of $x \cos(y/x) \frac{dy}{dx} = y \cos(y/x) + x$.
27.Solve: $(x^3+y^3) dy - x^2 y dx = 0$.
28.Solve: $y dx + x \log(y/x) dy - 2x dy = 0$.
29.Show that the DE $\frac{dy}{dx} = \frac{y \cos(y/x) + x \sin(y/x)}{x \cos(y/x)}$ is homogeneous and solve.
30.Solve: $(x \sqrt{x^2+y^2} - y^2) dx + xy dy = 0$.
31.Solve: $x \frac{dy}{dx} - y + x \sin(y/x) = 0$.
32.Find the particular solution of $(x^2+xy) dy = (x^2+y^2) dx$ given $y=1$ when $x=1$.
33.Solve: $2y e^{x/y} dx + (y - 2x e^{x/y}) dy = 0$.
34.Show that the solution of $\frac{dy}{dx} = \frac{x+2y}{x}$ is $x+y = cx^2$.
35.Solve: $\frac{dy}{dx} = \frac{xy}{x^2+y^2}$.
36.Find the general solution of $\frac{dy}{dx} + y \sec x = \tan x$.
37.Solve: $x \frac{dy}{dx} + 2y = x^2 \log x$.
38.Find the Integrating Factor of $(1-x^2) \frac{dy}{dx} - xy = 1$.
39.Solve: $\frac{dy}{dx} + \frac{2x}{1+x^2} y = \frac{4x^2}{1+x^2}$.
40.Find the particular solution of $\frac{dy}{dx} - 3y \cot x = \sin 2x$ given $y=2$ when $x=\pi/2$.
41.Solve: $(x+y+1) \frac{dy}{dx} = 1$. (Hint: Linear in $x$).
42.Find the general solution of $y dx - (x+2y^2) dy = 0$.
43.Solve: $\frac{dy}{dx} + y = \cos x - \sin x$.
44.Solve: $x \log x \frac{dy}{dx} + y = \frac{2}{x} \log x$.
45.Solve: $(1+x^2) dy + 2xy dx = \cot x dx$.
46.Find the particular solution of $(1+e^x) dy + (1+y^2) e^x dx = 0$ given $y=1$ when $x=0$. (Reducible case).
47.Solve: $\frac{dx}{dy} + \frac{x}{y} = y^2$.
48.Solve: $(x+2y^3) \frac{dy}{dx} = y$.
49.Find the general solution of $\cos^2 x \frac{dy}{dx} + y = \tan x$.
50.Solve the DE: $x \frac{dy}{dx} - y = (x-1) e^x$.
51.Solve: $\frac{dy}{dx} + \frac{y}{x} = y^2 \log x$. (Bernoulli's Equation).
52.Find the orthogonal trajectories of the family of curves $y = ax^2$.
53.The rate of growth of bacteria is proportional to the number present. If the population doubles in 5 hours, in how many hours will it triple?
54.A cup of tea at $100^\circ C$ is placed in a room at $20^\circ C$. If it cools to $60^\circ C$ in 10 minutes, find its temperature after 20 minutes.
55.Find the equation of the curve passing through $(0,-2)$ given that at any point $(x,y)$ on the curve, the product of the slope of its tangent and y-coordinate is equal to the x-coordinate.
56.Solve: $x \frac{dy}{dx} + y = y^2 \log x$.
57.Find the orthogonal trajectories of $x^2 + y^2 = c^2$.
58.In a bank, principal increases continuously at the rate of 5% per year. In how many years will Rs 1000 double itself?
59.Solve: $\frac{dy}{dx} - y \tan x = -y^2 \sec x$.
60.Find the equation of the curve which passes through the point $(1,1)$ and whose slope is $\frac{2y}{x}$.
