Summary and Quick Revision

The Chapter in One Sweep

A differential equation relates a function to its derivatives. This chapter teaches you to do exactly two things: read such an equation (order, degree, what counts as a solution) and solve the three first-order types that boards and JEE actually ask.

Reading an equation

Order = the order of the highest derivative present.
Degree = the power of that highest-order derivative, but only after the equation is polynomial in its derivatives — clear roots and fractions first; if a derivative sits inside $\sin$ , $e^{(\cdot)}$ , $\log$ , the degree is not defined. (A function like $\sin y$ wrapping the variable is harmless.)

$Decision chart for reading order and degree of a differential equation$

General vs particular: the general solution of an order- $n$ equation carries exactly $n$ arbitrary constants; a particular solution carries none — the constants are fixed by given conditions. To verify a claimed solution, differentiate it and substitute; never try to re-solve.

$Family of curves for a general solution with one particular curve highlighted$

The three-method decision tree

Ask in this order:

Does the right side factor as $g(x)\,h(y)$ ? → Variables separable. Move all $y$ to $dy$ , all $x$ to $dx$ , integrate both sides, one constant.

$Four move pipeline for solving a separable differential equation$

Is every term the same total degree — is $\dfrac{dy}{dx}$ a function of $\dfrac{y}{x}$ alone? → Homogeneous. Substitute $y = vx$ , so $\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$ ; the equation in $v, x$ separates. (Mirror: for $\dfrac{dx}{dy} = g\left(\dfrac{x}{y}\right)$ use $x = vy$ .)

$Pipeline for solving a homogeneous equation with the y equals vx substitution$

Is it shaped $\dfrac{dy}{dx} + Py = Q$ with $P, Q$ functions of $x$ ? → Linear. Compute I.F. $= e^{\int P\,dx}$ , then $y \cdot \text{I.F.} = \displaystyle\int Q \cdot \text{I.F.}\,dx + C$ . (Mirror: $\dfrac{dx}{dy} + P_1 x = Q_1$ gives $x \cdot e^{\int P_1 dy} = \displaystyle\int Q_1 e^{\int P_1 dy}\,dy + C$ .)

$Pipeline for solving a linear differential equation with an integrating factor$

Tables and the Mistake Checklist

Integrating-factor speed table

$P(x)$	I.F. $= e^{\int P\,dx}$
$\dfrac{a}{x}$	$x^a$
$\tan x$	$\sec x$
$\cot x$	$\sin x$
$\sec x$	$\sec x + \tan x$
$-x$	$e^{-x^2/2}$
$\dfrac{1}{1 + y^2}$ (in $y$ )	$e^{\tan^{-1} y}$

Exact differentials worth recognising

Combination	Equals
$x\,dy + y\,dx$	$d(xy)$
$\dfrac{x\,dy - y\,dx}{x^2}$	$d\left(\dfrac{y}{x}\right)$
$\dfrac{y\,dx - x\,dy}{y^2}$	$d\left(\dfrac{x}{y}\right)$
$\dfrac{x\,dy - y\,dx}{x^2 + y^2}$	$d\left(\tan^{-1}\dfrac{y}{x}\right)$
$x\,dx + y\,dy$	$d\left(\dfrac{x^2 + y^2}{2}\right)$

Growth and decay in one line

$\dfrac{dy}{dt} = ky \Rightarrow y = y_0 e^{kt}$ — doubling time $\dfrac{\log 2}{k}$ ; each equal time interval multiplies $y$ by the same factor.

The mistake checklist — run it before submitting

Degree read before clearing radicals? Square away fractional powers first; degree must be a positive integer or "not defined".
Constant added at the moment of integration? One $C$ , on the $x$ side, immediately — not appended at the end.
Logs merged before naming $C$ ? $\log|y| = \log|x| + C$ should become $y = Cx$ by exponentiating properly — a dangling un-merged constant changes the family.
I.F. multiplied into $Q$ as well? The solution line is $y\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx$ — forgetting the I.F. inside the integral is the top linear-equation error.
Returned from $v$ to $\dfrac{y}{x}$ (or $\dfrac{1}{y}$ in Bernoulli)? The answer must be in $x$ and $y$ ; and when multiplying back through, the constant picks up factors too.
Particular solution actually particular? Substitute the condition, solve for $C$ , and state the final constant-free equation.
Verified by substitution? Thirty seconds: differentiate your answer and push it through the original equation.

Where each skill was built

Order and degree — Section 1. Verification and families — Section 2. Separable equations and growth models — Section 3. Homogeneous equations — Section 4. Linear equations — Section 5. Graded worked examples — Section 6. Board-pattern practice — Section 7. JEE tools (shift substitution, Bernoulli, exact differentials, orthogonal trajectories) — Section 8. Full JEE drill — Section 9.

Differential Equations