CBSE Class 12 Maths: Differential Equations — Notes 2026

Order and Degree

• Order: highest order derivative in DE; degree: power of highest order derivative (when free from radicals)
• Example: (d²y/dx²)³ + dy/dx = 0; order = 2, degree = 3
• Degree not defined if DE involves sin, cos, log of derivatives (transcendental functions)

Formation of Differential Equations

• Differentiate the given relation as many times as there are arbitrary constants; eliminate constants
• Family of circles x² + y² = r²: differentiate → x + y(dy/dx) = 0; DE represents all circles centred at origin
• Eliminating two constants requires two differentiations; general solution has n constants for nth order DE

Variable Separable Method

• Separate variables: f(y)dy = g(x)dx; integrate both sides; add constant of integration C only once
• Example: dy/dx = xy → dy/y = x dx → ln|y| = x²/2 + C → y = Ae^(x²/2)
• Check: substitute back to verify solution satisfies original DE

Homogeneous Differential Equations

• Homogeneous: dy/dx = f(y/x); substitute y = vx → dy/dx = v + x(dv/dx)
• After substitution: variable separable in v and x; solve; back-substitute y = vx at end
• Recognising homogeneous: f(tx, ty) = f(x,y) for all t (same degree in numerator and denominator)

Linear Differential Equations

• Standard form: dy/dx + P(x)y = Q(x); P and Q functions of x only
• Integrating factor (IF): IF = e^(∫P dx); multiply both sides by IF
• Solution: y × IF = ∫Q × IF dx + C; general solution form

Applications of DE

• Population growth: dP/dt = rP; solution P = P₀ eʳᵗ; exponential growth model
• Newton's law of cooling: dT/dt = −k(T−T_env); solution T = T_env + (T₀−T_env) e^(−kt)
• Growth/decay problems: find k from initial conditions; use to find value at other times

CBSE Board Focus

• Differential Equations: 5–7 marks; variable separable and linear DE most tested
• Formation of DE: if given family of curves, form DE by eliminating arbitrary constants
• Particular solution: when initial condition given (y = y₀ when x = x₀); find C and write specific solution