Differential Equations
Differential Equations
The exam tests order and degree, formation of differential equations, general and particular solutions, and solving variable-separable, homogeneous and first-order linear equations.
Core rules
- Order is the highest derivative present; degree is the power of the highest derivative when the equation is polynomial in derivatives.
- A general solution contains as many arbitrary constants as the order; a particular solution fixes them using initial conditions.
- Variable separable: write dy/dx = f(x)g(y), separate, and integrate both sides.
- Homogeneous equations: substitute y = vx, so dy/dx = v + x dv/dx, then separate.
- Linear equation dy/dx + Py = Q has integrating factor IF = e^∫P dx, and the solution is y·IF = ∫Q·IF dx.
Worked example
Solve dy/dx + y/x = x², x > 0.
Here P = 1/x and Q = x², so IF = e^∫(1/x)dx = e^ln x = x. Multiplying through gives x dy/dx + y = x³, that is, d/dx(xy) = x³. Integrating: xy = x⁴/4 + C, so y = x³/4 + C/x.
Traps
- Defining degree when the equation is not polynomial in derivatives.
- Dividing by y and losing the constant solution y = 0.
- Using the wrong sign or exponent in the integrating factor.
- Forgetting to multiply Q by the integrating factor as well.
- Treating a homogeneous equation as linear, or vice versa.