Differential Equations
Questions test order and degree, formation of a differential equation from a family of curves, and solution methods: variable separable, homogeneous, and linear equations with an integrating factor. Expect a 5–6 mark solve-and-find-particular-solution problem.
Definitions and methods
- Order = highest derivative present; degree = power of the highest derivative after the equation is made polynomial in derivatives.
- Variable separable: bring to f(y) dy = g(x) dx and integrate both sides.
- Homogeneous: substitute y = vx, so dy/dx = v + x dv/dx.
- Linear form dy/dx + Py = Q: integrating factor IF = e^(∫P dx), and the solution is y·IF = ∫Q·IF dx.
- A first-order equation needs one arbitrary constant; use the initial condition to fix it.
Worked example
Solve dy/dx + y/x = x². Here P = 1/x and Q = x², so IF = e^(∫dx/x) = e^(log x) = x. Multiplying through, x dy/dx + y = x³, i.e. d/dx (xy) = x³. Integrating, xy = x⁴/4 + C, so y = x³/4 + C/x.
Traps
- Stating the degree when the equation is not polynomial in derivatives; degree is then not defined.
- Writing the integrating factor as e^(∫P dy).
- Forgetting the arbitrary constant, or adding two constants for a first-order equation.
- Losing the modulus in e^(log|x|); take the positive branch for the integrating factor.