Differential Equations
Differential Equations examines equations containing derivatives. The exam tests order and degree, formation of equations, solution by separation of variables, the first-order linear form, and simple growth or decay applications.
Core rules
- Order is the highest derivative present. Degree is the power of the highest-order derivative after the equation is cleared of radicals and fractions in derivatives.
- A general solution contains as many arbitrary constants as the order; a particular solution uses given conditions to fix them.
- Variable separable: rewrite dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, then integrate both sides.
- Linear first-order form: dy/dx + Py = Q, where P and Q are functions of x. The integrating factor is e^{∫P dx}, and the solution is y(IF) = ∫Q(IF) dx + C.
- Growth and decay models use dy/dt = ky, giving y = y₀e^{kt}; k > 0 indicates growth and k < 0 decay.
Worked example
Solve dy/dx = 2xy with y(0) = 1. Separate: dy/y = 2x dx. Integrate: ln|y| = x² + C, so y = Ae^{x²}. Apply y(0) = 1: 1 = Ae⁰, so A = 1. Particular solution: y = e^{x²}. Check: dy/dx = 2x e^{x²} = 2xy, and y(0) = 1.
Traps
Finding degree before removing a square root or fraction in the derivative; losing the constant of integration; writing the integrating factor as e^{∫Q dx} instead of e^{∫P dx}; and substituting the initial condition before integrating.