Differential Equations
Differential-equation practice develops the ability to 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.
- For continuously differentiable coefficients on a suitable simply connected domain, ∂M/∂y = ∂N/∂x gives exactness of M dx + N dy; find a potential F with F_x = M and F_y = N. Locally, check the corresponding neighborhood assumptions.
- 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 trial duplicates a complementary solution, multiply by a power of x equal to the relevant characteristic-root multiplicity.
- 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.