Differential Equations

A differential equation (DE) is an equation involving derivatives of an unknown function.

Order = highest derivative present.
Degree = power of the highest-order derivative (after rationalizing).

Example: (dy/dx)³ + 5y = 0 — order 1, degree 3.

Method 1: Variable separable.
If dy/dx can be written as f(x)·g(y), separate variables:

dy / g(y) = f(x) dx → integrate both sides.

Example: dy/dx = y x²
→ dy/y = x² dx
→ ln|y| = x³/3 + C
→ y = C · e^(x³/3)

Method 2: Homogeneous DE.
If dy/dx = F(x, y) where F is a function of y/x only:

Substitute v = y/x (so y = vx, dy/dx = v + x dv/dx). The equation reduces to variable separable in v and x.

Example: dy/dx = (x + y)/x = 1 + y/x. Let v = y/x:
v + x dv/dx = 1 + v → x dv/dx = 1 → dv = dx/x → v = ln|x| + C
→ y = x(ln|x| + C)

Method 3: Linear DE (first order).
Form: dy/dx + P(x) y = Q(x).

Integrating factor (IF): μ(x) = e^∫P(x)dx.

Multiplying through by μ makes the LHS = d/dx[μy], so:

μy = ∫ μ Q(x) dx + C

Example: dy/dx + y = e^x. Here P=1, Q=e^x. IF = e^x.
d/dx[e^x · y] = e^x · e^x = e^(2x)
e^x y = e^(2x)/2 + C
→ y = e^x/2 + C·e^(−x)

Method 4: Exact DE.
M(x,y) dx + N(x,y) dy = 0 is exact if ∂M/∂y = ∂N/∂x. Then exists F with dF = M dx + N dy → F = C.

If not exact, sometimes a multiplier (integrating factor) makes it exact.

Second-order linear DE with constant coefficients:

a y′′ + b y′ + c y = f(x).

Solution = homogeneous solution + particular solution.

Homogeneous: solve characteristic equation a m² + b m + c = 0.

• Two distinct real roots m₁, m₂: y_h = A e^(m₁x) + B e^(m₂x).
• Repeated root m: y_h = (A + Bx) e^(mx).
• Complex roots α ± iβ: y_h = e^(αx) (A cos βx + B sin βx).

Common JEE-style ODE applications:

• Population growth: dN/dt = kN → N = N₀ e^(kt).
• Newton's law of cooling: dT/dt = −k(T − T_room) → T − T_room = (T₀ − T_room) e^(−kt).
• Radioactive decay: dN/dt = −λN → N = N₀ e^(−λt).
• RC circuit charging: V_C(t) = V₀ (1 − e^(−t/RC)).