Pattern 1: Linear DE with IF (Most Frequent) JEE gives dy/dx + P(x)y = Q(x) with a specific IF. Common IFs: $x^n$, sec x, e^($x^2$), (1+$x^2$). The solution form yIF = integral QIF dx + C is applied with an initial condition to find C.

Pattern 2: Homogeneous DE (High Frequency) Given dy/dx = $\frac{ax+by}{(cx+dy)}$, recognize homogeneity and use y = vx. JEE often asks for the curve passing through a specific point.

Pattern 3: Formation of DE (Moderate) Given a 1- or 2-parameter family, find the DE. Key: count constants to determine order, then differentiate and eliminate.

Pattern 4: Bernoulli Equation (Moderate) dy/dx + Py = $Qy^n$ typically with n=2 or n=-1. Recognize the $y^n$ term on the right side.

Pattern 5: Exact Differentials (Moderate) Recognizing d(xy), d$\frac{y}{x}$, d(arctan$\frac{y}{x}$) in disguised forms. JEE loves x dy - y dx expressions.

Pattern 6: Application Problems (Moderate) Growth/decay, cooling, or geometric properties (subnormal, subtangent). These translate to standard DE types.

Time-saving tips:

• Check separable first (quickest)
• For dy/dx = $\frac{linear in x,y}{(linear in x,y)}$: check homogeneity, then try linear
• If $y^n$ appears on RHS, it's Bernoulli
• x dy + y dx or x dy - y dx signals exact differential
• Always verify by substituting back if time permits