Differential Equations
Differential equations in the IPE papers test order and degree, formation of a differential equation by eliminating arbitrary constants, and the three standard solution methods: variables separable, homogeneous substitution, and the first-order linear equation. Applications to growth and decay also appear.
Core rules and concepts
- Order is the highest derivative present; degree is the power of the highest derivative after the equation is made polynomial in derivatives.
- Formation: differentiate as many times as there are arbitrary constants, then eliminate them.
- Variable separable: write dy/dx = f(x)g(y) and integrate ∫dy/g(y) = ∫f(x) dx.
- Homogeneous equation: substitute y = vx, so dy/dx = v + x dv/dx, then separate.
- Linear equation dy/dx + Py = Q, with P and Q functions of x: integrating factor IF = e^∫P dx, solution y·IF = ∫Q·IF dx.
Worked example
Solve dy/dx + y/x = x². Here P = 1/x and Q = x², so IF = e^∫(1/x)dx = e^{ln x} = x. Multiplying through: x dy/dx + y = x³, that is d(xy)/dx = x³. Integrating: xy = x⁴/4 + C, so y = x³/4 + C/x.
Traps that cost marks
- Quoting a degree when the equation involves sin(dy/dx) or e^{dy/dx}; the degree is then undefined.
- Using the wrong sign for P when building the integrating factor.
- Forgetting that the homogeneous substitution also replaces dy/dx, not just y.
- Losing the solution y = 0 after dividing by a factor of y.