Differential Equations
Differential equations on this paper test whether you can classify an equation, choose the right solution method, and apply initial or boundary conditions without losing constants along the way.
First order
- Separable: separate and integrate, then apply the initial condition before rearranging if possible.
- Linear y' + P(x)y = Q(x): integrating factor μ = exp(∫P dx), giving (μy)' = μQ.
- Exact equations M dx + N dy = 0 satisfy ∂M/∂y = ∂N/∂x; find a potential F with F_x = M and F_y = N.
- Equilibria of y' = f(y) occur where f(y) = 0; the sign of f between equilibria gives stability on the phase line.
Second order linear, constant coefficients
- Homogeneous: solve ar² + br + c = 0. Distinct real roots give Ae^{r₁x} + Be^{r₂x}; a repeated root gives (A + Bx)e^{rx}; complex α ± iβ gives e^{αx}(A cos βx + B sin βx).
- Particular integral by undetermined coefficients; if the guess duplicates a complementary solution, multiply by x.
- General solution = complementary + particular; apply initial conditions last.
Worked example
Solve y' − y = e^{2x}, y(0) = 3. Here P = −1, so μ = e^{−x}. Multiply through: e^{−x}y' − e^{−x}y = e^{x}, i.e. (e^{−x}y)' = e^{x}. Integrate: e^{−x}y = e^{x} + C, so y = e^{2x} + Ce^{x}. Apply y(0) = 3: 1 + C = 3, so C = 2 and y = e^{2x} + 2e^{x}. Check: y' = 2e^{2x} + 2e^{x}, so y' − y = e^{2x}. Consistent.
Traps that cost marks
- Sign error in the integrating factor: for y' − y = Q, P = −1, not +1.
- Forgetting the x multiplier in the repeated-root case or when the forcing term resonates.
- Applying the initial condition to the complementary function only.
- Dividing by a variable that could be zero and losing a singular solution.
- Writing x where the characteristic variable r belongs.