1.Answer: Order (of the differential equation).
2.Answer: Order is 2 (since the highest derivative is $\frac{d^2y}{dx^2}$).
3.Answer: Order is 1 (the highest derivative is $\frac{dy}{dx}$).
4.Answer: Degree is 1 (the power of the highest order derivative $\frac{d^2y}{dx^2}$ is 1).
5.Answer: No. (Because the derivative is inside a trigonometric function, it cannot be written as a polynomial equation in derivatives).
6.Answer: No. (Unless it can be simplified/inverted cleanly into a polynomial form without trapping other derivatives).
7.Answer: $n$ (The number of arbitrary constants equals the order).
8.Answer: Zero (0). Particular solutions have no arbitrary constants.
9.Answer: Order is 2 (Since there are two arbitrary constants, $A$ and $B$).
10.Answer: Square both sides: $\frac{dy}{dx} = (x+y)^2$. Order is 1, Degree is 1.
11.Answer: Twice (2 times).
12.Answer: $\frac{dy}{dx} = c$.
13.Answer: Group all $y$ terms (with $dy$) on one side, and all $x$ terms (with $dx$) on the other side.
14.Answer: $y \,dy = x \,dx$.
15.Answer: $\int y \,dy = \int x \,dx \implies \frac{y^2}{2} = \frac{x^2}{2} + C$.
16.Answer: $\frac{dy}{y} = x^2 \,dx$.
17.Answer: $\frac{dy}{dx} = e^x \cdot e^y \implies \frac{dy}{e^y} = e^x \,dx \implies e^{-y} \,dy = e^x \,dx$.
18.Answer: $\int \frac{1}{y} \,dy = \int \frac{1}{x} \,dx \implies \ln|y| = \ln|x| + C$.
19.Answer: Let $ax + by + c = t$.
20.Answer: Divide by $\tan x \tan y$: $\frac{\sec^2 x}{\tan x} dx + \frac{\sec^2 y}{\tan y} dy = 0$.
22.Answer: Degree 2. ($F(\lambda x, \lambda y) = \lambda^2 x^2 + \lambda^2 y^2 = \lambda^2 (x^2+y^2)$).
23.Answer: No. (The degrees of the terms don't match; $\lambda$ gets trapped inside the sine function).
24.Answer: Substitute $y = vx$ (which means $v = y/x$).
25.Answer: $\frac{dy}{dx} = v + x \frac{dv}{dx}$.
26.Answer: Substitute $x = vy$ (which means $v = x/y$).
27.Answer: $\frac{dx}{dy} = v + y \frac{dv}{dy}$.
28.Answer: Yes. (Both numerator and denominator have terms of degree 1. Dividing by $x$ gives $1 + (y/x)$, degree 0).
29.Answer: $\frac{x + vx}{x} = \frac{x(1+v)}{x} = 1 + v$.
30.Answer: Yes. ($F(\lambda x, \lambda y) = e^{\lambda y / \lambda x} = e^{y/x} = \lambda^0 F(x,y)$).
31.Answer: $\frac{dy}{dx} + Py = Q$.
32.Answer: Functions of $x$ alone (or constants).
33.Answer: Integrating Factor.
34.Answer: I.F. = $e^{\int P \,dx}$.
35.Answer: $P = 2$ and $Q = \sin x$.
36.Answer: I.F. = $e^{\int 2 \,dx} = e^{2x}$.
37.Answer: $P = \frac{1}{x}$.
38.Answer: I.F. = $e^{\int \frac{1}{x} \,dx} = e^{\ln x} = x$.
39.Answer: $y \cdot (\text{I.F.}) = \int [Q \cdot (\text{I.F.})] \,dx + C$.
40.Answer: $\frac{dx}{dy} + P_1 x = Q_1$ (where $P_1, Q_1$ are functions of $y$).
41.Answer: $\frac{dy}{dx} + Py = Qy^n$ (where $n \neq 0, 1$).
42.Answer: Divide by $y^n$.
43.Answer: Substitute $v = y^{1-n}$.
44.Answer: $\frac{dP}{dt} = kP$ (where $k$ is the proportionality constant).
45.Answer: $\frac{dP}{P} = k \,dt$.
46.Answer: $\ln|P| = kt + C \implies P = C_1 e^{kt}$.
47.Answer: The difference between the temperature of the object and the temperature of the surroundings ($T - T_s$).
48.Answer: $\frac{dT}{dt} = -k(T - T_s)$ (where $k > 0$).
49.Answer: $\frac{dy}{dx} = 2x$.
50.Answer: $y = x^2 + C$ (a family of parabolas).
