Solving differential equations

The essentials

Three types of first-order equations are solved in Class 12.

Variable separable: dy/dx = g(x)h(y). Rearrange to dy/h(y) = g(x) dx and integrate both sides. Example: dy/dx = y gives log|y| = x + c, so y = Ceˣ.

Homogeneous: dy/dx = F(x, y) where F(λx, λy) = F(x, y), so the right side depends only on y/x. Put y = vx, so dy/dx = v + x dv/dx, and the equation becomes separable in v and x.

Linear: dy/dx + Py = Q, where P and Q are functions of x only. Multiply by the integrating factor IF = e^(∫P dx). Then the solution is y × IF = ∫ (Q × IF) dx + C.

Key points

• For dy/dx + 2y = eˣ, P = 2, so IF = e^(2x).
• For dy/dx + y/x = x² (x > 0), IF = e^(log x) = x, and the left side becomes d(xy)/dx.
• An initial condition fixes the constant and gives the particular solution.
• A function F(x, y) is homogeneous of degree n if F(λx, λy) = λⁿF(x, y); dy/dx = F works with the y = vx method when F has degree zero.

Exam tips

• First identify the type; the method follows automatically.
• Write the linear form with coefficient of dy/dx equal to 1 before reading off P.

Common mistakes

• Forgetting the constant of integration until the end, then being unable to apply the initial condition.
• Taking IF = ∫P dx instead of e^(∫P dx).