Chapter 1: Variable Separable The simplest type. If dy/dx = f(x)*g(y), separate: dy/g(y) = f(x)dx. Integrate both sides. Example: dy/dx = xy gives dy/y = x dx, so ln|y| = $x^2$/2 + C. Variations include reducible separable: dy/dx = f(ax+by+c) where substitution v = ax+by+c works.

Chapter 2: Homogeneous Equations dy/dx = f$\frac{x,y}{g}$(x,y) where f,g are homogeneous of the same degree. Test: replace (x,y) by (tx,ty) — if t cancels, it's homogeneous. Substitute y = vx: dy/dx = v + xv', reducing to separable form. For $\frac{ax+by+c}{(dx+ey+f)}$ type with c,f != 0: if a/d = $\frac{b}{e}$, use v = ax+by; otherwise translate to remove constants.

Chapter 3: Linear First-Order DE The most important type for JEE. Standard form dy/dx + P(x)y = Q(x). Multiply by IF = e^(integral P dx). Common IFs: P = $\frac{k}{x}$ gives IF = $x^k$; P = k gives IF = e^(kx); P = tan(x) gives IF = sec(x); P = cot(x) gives IF = sin(x).

Trick: If the equation looks linear in x (not y), rewrite as dx/dy + P(y)x = Q(y) and find IF with respect to y.

Chapter 4: Bernoulli's Equation dy/dx + Py = $Qy^n$. Divide by $y^n$, substitute v = y^(1-n). The resulting linear equation is dv/dx + (1-n)Pv = (1-n)Q. Most common in JEE: n = 2 (substitute v = 1/y) and n = -1 (substitute v = $y^2$).

Chapter 5: Exact Equations and Integrating Factors For M dx + N dy = 0: check dM/dy = $\frac{dN}{dx}$. If exact, find F by integrating M w.r.t. x, then match dF/dy with N. In JEE, recognizing exact differentials (d(xy), d$\frac{y}{x}$, etc.) is more practical than the general method.

Chapter 6: Formation and Applications Formation: given family with n constants, differentiate n times, eliminate constants. The resulting DE has order n. Applications: growth/decay, cooling, mixture problems, orthogonal trajectories. For orthogonal trajectories, find the DE of the given family, replace dy/dx by -dx/dy, and solve.