GATE MA: Ordinary Differential Equations

Ordinary Differential Equations in GATE MA emphasizes solution techniques, existence–uniqueness, and qualitative behavior. Expect first-order equations, constant-coefficient linear systems, Laplace transforms, and Sturm–Liouville theory.

Core rules

• First-order linear: y′+P(x)y=Q(x); integrating factor μ=e^{∫P dx}.
• Exact equation M dx+N dy=0 requires M_y=N_x; then find potential F with F_x=M, F_y=N.
• Second-order linear: y″+p y′+q y=0; Wronskian W=y_1y_2′−y_1′y_2. Solutions are independent iff W≠0.
• Constant coefficients: solve the characteristic equation; resonance occurs when the forcing matches a homogeneous solution.
• Picard–Lindelöf: y′=f(x,y), y(x_0)=y_0 has a unique local solution if f and ∂f/∂y are continuous.
• Bernoulli equation y′+P y=Q y^n uses v=y^{1−n}.

Worked example

Solve y′+2y=e^{−x}, y(0)=1. Integrating factor μ=e^{2x}. Then (e^{2x}y)′=e^{x}, so e^{2x}y=e^x+C and y=e^{−x}+Ce^{−2x}. Using y(0)=1 gives 1=1+C, so C=0; y=e^{−x}.

Traps

• The Wronskian can vanish at a point without the functions being dependent; on an interval, nonzero W implies independence.
• For y″+y=sin x, the forcing resonates; the particular solution needs an extra factor x.
• Laplace transforms require initial conditions at 0; a shift changes the forcing term.
• Singular points of series solutions need classification as regular or irregular before using Frobenius.