Chapter 9: Differential Equations

A differential equation relates a function to its derivatives. Examples are everywhere: Newton's second law $m\dfrac{d^2x}{dt^2} = F(x, t)$, the radioactive decay law $\dfrac{dN}{dt} = -kN$, the cooling law of Newton $\dfrac{dT}{dt} = -k(T - T_{\text{env}})$. To solve a differential equation means to find the function (or family of functions) that satisfy it. The toolkit you developed in Chapter 7 , integration , is now used systematically to recover unknown functions from information about their rates of change.

The first task is taxonomy. Each differential equation has an order (the highest derivative appearing) and a degree (the power of the highest-order derivative, after clearing radicals and fractions). The order determines how many constants of integration appear in the general solution , and therefore how many initial conditions you need to pin down a particular solution.

The Class XII syllabus restricts to first-order equations. Within those, three types yield to elementary methods. Variable-separable equations look like $\dfrac{dy}{dx} = f(x)\,g(y)$: rearrange to separate $x$'s and $y$'s, and integrate each side. Homogeneous equations have $\dfrac{dy}{dx} = F(y/x)$; the substitution $v = y/x$ converts them to separable form. Linear equations have the form $\dfrac{dy}{dx} + P(x)y = Q(x)$; multiplying by the integrating factor $e^{\int P\,dx}$ turns the left side into a perfect derivative.

The chapter has obvious physical relevance. The differential equations of growth, decay, mixing, motion under resistance , all are first-order and solvable by the methods above. JEE problems often dress up a physics setting (a population, a leaking tank, an RL circuit) and ask you to model and solve.

The skill you must develop is classification. Given an equation, decide quickly: separable? homogeneous? linear? Once classified, the method is mechanical. Misclassification leads to wasted effort.

Prerequisites: comfortable integration (especially partial fractions and substitution); familiarity with logarithms; and the chain rule. Initial conditions translate algebraic constraints onto the constant of integration, so attention to algebra matters.

What's inside

1. Order, degree, and the meaning of a solution , taxonomy of differential equations.
2. Forming a differential equation , given a family of curves, find the DE it satisfies.
3. Variable-separable equations , separate, integrate, combine constants.
4. Homogeneous differential equations , substitute $y = vx$.
5. Linear differential equations and integrating factor , the workhorse method.
6. Initial-value problems and applications , translate word problems into DEs.

Key results / Formula card

How to read this chapter

Classification is everything. After reading each topic, build a flowchart: see $\dfrac{dy}{dx} = f(x)g(y)$? Separable. See a homogeneous expression of $x$ and $y$? Substitute $y = vx$. See $\dfrac{dy}{dx} + P y = Q$? Integrating factor. With this template internalised, you can solve any board-level differential equation in minutes.