Differential equations (DEs) form a critical topic in JEE Main, carrying 2-3 questions per year. A differential equation is an equation involving an unknown function and its derivatives. The order is the highest derivative's order, and the degree is the power of that highest derivative (when the equation is polynomial in derivatives).

Formation of DEs starts from a family of curves with n arbitrary constants. Differentiate n times and eliminate the constants to obtain a DE of order n. For example, y = Ae^(2x) + Be^(-x) has 2 constants, giving a second-order DE: y'' - y' - 2y = 0.

The classification of first-order DEs determines the solution method. Variable Separable equations can be written as f(x)dx = g(y)dy and solved by direct integration. Homogeneous DEs satisfy dy/dx = g$\frac{y}{x}$ and are solved by the substitution y = vx, which reduces them to separable form. Linear DEs have the form dy/dx + P(x)y = Q(x) and are solved using the integrating factor IF = e^(integral P dx), giving yIF = integral QIF dx + C.

Bernoulli's equation dy/dx + Py = $Qy^n$ (n != 0,1) is converted to linear form by substituting v = y^(1-n). Exact DEs M dx + N dy = 0 satisfy dM/dy = $\frac{dN}{dx}$, and their solution is found by integrating M with respect to x and N with respect to y.

Recognizing exact differentials is a powerful shortcut: d(xy) = x dy + y dx, d$\frac{y}{x}$ = $\frac{x dy - y dx}{x}$^2, d(arctan$\frac{y}{x}$) = $\frac{x dy - y dx}{(x^2+y^2)}$. These appear frequently in JEE problems disguised in complex forms.

Clairaut's equation y = xy' + f(y') has general solution y = cx + f(c) and a singular solution obtained by eliminating c between y = cx + f(c) and 0 = x + f'(c).

Applications include exponential growth/decay (dN/dt = kN, giving N = $N_0$ e^(kt)), Newton's law of cooling (dT/dt = -k(T-$T_0$)), and orthogonal trajectories (replace dy/dx by -dx/dy in the family's DE).

The key problem-solving strategy is: first identify the type (separable > homogeneous > Bernoulli/linear > exact), then apply the standard method. For equations of the form dy/dx = $\frac{ax+by+c}{(dx+ey+f)}$ where ad != be, translate origin to eliminate constants; if ad = be, substitute v = ax+by.