Differential Equations

Syllabus Unit: Engineering Mathematics

Why it matters

Chapter overview

Differential equations translate physical laws — rates of decay, circuit currents, vibrating systems — into solvable algebra; GATE's Engineering Mathematics section tests first-order equations (separable, linear with integrating factor) and higher-order linear ODEs with constant coefficients, plus their direct application to RC/RL/RLC circuits and spring-mass-damper systems.

General Aptitude & Mathematics Engineering Mathematics 15% weightage

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Identify dy/dx = f(x)g(y)

Separate: dy/g(y) = f(x)dx

Integrate both sides independently

Apply the initial condition for the particular solution

1First-Order ODEs & Separable Equations

A separable ODE lets you herd every x to one side and every y to the other — then it's just two ordinary integrals.

Learning Objectives

Recognise a separable ODE dy/dx = f(x)g(y) and rewrite it as dy/g(y) = f(x)dx.
Solve a separable ODE by integrating both sides and applying an initial condition.
Solve the standard growth/decay ODE dy/dx = ky and interpret k's sign physically.
Identify and solve homogeneous ODEs dy/dx = f(y/x) using the substitution v = y/x.

Key Terms

Separable ODEAn ODE of the form dy/dx = f(x)g(y), solvable by moving all y-terms and all x-terms to opposite sides before integrating.

General SolutionThe full family of solution curves to an ODE, containing an arbitrary constant.

Particular SolutionOne specific curve from the general solution, obtained by fixing the constant using a given initial condition.

Homogeneous ODE (degree-0)An ODE where dy/dx can be written purely as a function of the ratio y/x, solved via the substitution y = vx.

A separable ODE dy/dx = f(x)g(y) is solved by algebraically separating the variables: ∫ dy/g(y) = ∫ f(x) dx + C — turning a differential equation into two ordinary integrals.

Worked pattern: exponential growth/decay

dy/dx = ky separates to dy/y = k dx, giving ln|y| = kx + C, so y = Ae^(kx). k>0 models growth (population, compound interest); k<0 models decay (radioactive decay, Newton's law of cooling).

Exam TipAn initial condition like y(0)=y0 pins down the arbitrary constant to A = y0 — GATE almost always supplies one, converting the general solution into a unique particular solution.

Homogeneous equations via v = y/x

If dy/dx can be written purely as a function of (y/x), substitute y = vx, so dy/dx = v + x(dv/dx); this converts the equation into a new one in v and x that is always separable.

Common MistakeDon't confuse this "homogeneous ODE" (degree-0 in y/x) with a "homogeneous linear ODE" (RHS = 0) from later topics — GATE reuses the same word for two different ideas.