Solving Differential Equations – IIT‑JEE (Main + Advanced)
A concise, self‑contained guide to the most common DE techniques that appear in the JEE syllabus.
1. Simple Explanation
A differential equation (DE) is a relation that involves an unknown function (y(x)) and its derivatives.
The goal is to find all functions (y(x)) that satisfy the relation.
In the JEE context you will see:
| Type | Typical form | What you need to know |
|---|---|---|
| First‑order | (y' = f(x,y)) or (P(x) y' + Q(x) y = R(x)) | Separable, Linear, Exact, Bernoulli, Integrating Factor |
| Second‑order (linear) | (y'' + p(x)y' + q(x)y = r(x)) | Homogeneous + Particular, Characteristic equation, Reduction of order |
| Higher‑order | Rare | Usually reducible to lower order |
| Non‑linear | ((y')^2 = f(x,y)) | Separable, Bernoulli, Homogeneous, or special substitutions |
Key idea – Reduce the DE to a form you can integrate directly, or use a known formula (characteristic equation, integrating factor, etc.).
2. Key Points (Bullet‑style Cheat Sheet)
| Category | Method | Conditions | Resulting Integral |
|---|---|---|---|
| First‑order | Separable | Can be written ( \frac{dy}{dx} = g(x)h(y) ) | (\displaystyle \int \frac{dy}{h(y)} = \int g(x),dx) |
| Linear | (y' + p(x)y = q(x)) | Integrating factor ( \mu(x)=e^{\int p(x)dx}) → (\displaystyle (\mu y)' = \mu q) | |
| Exact | (M(x,y)dx+N(x,y)dy=0) with (M_y=N_x) | (\displaystyle \Phi(x,y)=C) where (\Phi_x=M, \Phi_y=N) | |
| Bernoulli | (y'+p(x)y = q(x)y^n) ( (n\ne0,1) ) | Substitute (v=y^{1-n}) → linear in (v) | |
| Homogeneous | (F(\frac{y}{x})) or (M(x,y)dx+N(x,y)dy=0) with (M,N) homogeneous | (v=\frac{y}{x}) → separable in (v,x) | |
| Second‑order linear | Homogeneous | (y''+p(x)y'+q(x)y=0) | Solve characteristic (m^2+ap+b=0) → (y = C_1e^{m_1x}+C_2e^{m_2x}) (or trig if complex) |
| Particular (und. coefficients) | RHS a known function | Assume form (polynomial, sine, exponential), plug‑in, solve for constants | |
| Method of Und. Coefficients | RHS is (P_n(x)e^{\lambda x}) or (P_n(x)\cos\beta x,, P_n(x)\sin\beta x) | Try (y_p = x^s Q_n(x)e^{\lambda x}) etc. | |
| Variation of Parameters | General RHS; less common in JEE | (y_p = u_1y_1+u_2y_2,\ u'_1y_1+u'_2y_2 = 0,\ u'_1y'_1+u'_2y'_2 = r(x)) | |
| Reduction of Order | One solution (y_1) known | Let (y = v(x)y_1) → first‑order for (v') |
Tip: Always check if the DE is homogeneous (in the sense of order) or exact. These are the first checks before trying more complicated methods.
3. Worked Example (JEE‑style)
Problem
Solve the differential equation
[ x^2y'' - 3xy' + 4y = 0 ]
with the initial conditions (y(1)=2) and (y'(1)=5).
Step 1: Identify the equation
It is a second‑order linear homogeneous ODE with variable coefficients of the Cauchy–Euler type.
Step 2: Use the standard substitution
For a Cauchy–Euler equation (x^2y'' + axy' + by = 0), try (y = x^m).
Compute derivatives:
[ y' = mx^{m-1},\qquad y'' = m(m-1)x^{m-2} ]
Plug into the ODE:
[ x^2[m(m-1)x^{m-2}] - 3x[mx^{m-1}] + 4x^m = 0 ]
Simplify:
[ m(m-1)x^m - 3mx^m + 4x^m = 0 ] [ [m(m-1) - 3m + 4]x^m = 0 ]
Thus the indicial (characteristic) equation is
[ m^2 - m - 3m + 4 = m^2 - 4m + 4 = 0 ] [ (m-2)^2 = 0 \quad \Longrightarrow \quad m = 2 \text{ (double root)} ]
Step 3: Write the general solution
For a repeated root (m),
[ y(x) = C_1x^m + C_2x^m \ln x = C_1x^2 + C_2x^2\ln x ]