Differential Equations
What is tested
Differential Equations asks for order and degree, formation of a differential equation from a family of curves, solution of variable-separable, homogeneous and first-order linear equations, and particular solutions fixed by an initial condition.
Core rules
- Order is the highest derivative present; degree is the power of that derivative once the equation is polynomial in derivatives.
- Variable separable: write f(y)dy = g(x)dx and integrate once, adding a single arbitrary constant.
- Homogeneous equation dy/dx = F(x,y) with F of degree zero: put y = vx, so dy/dx = v + x dv/dx, then separate.
- Linear equation dy/dx + Py = Q: integrating factor IF = e^∫P dx, and the solution is y·IF = ∫Q·IF dx.
- A particular solution uses the given condition to evaluate C.
Worked example
Solve dy/dx + (2/x)y = x² for x > 0. Here P = 2/x, so IF = e^∫(2/x)dx = e^(2 log x) = x². Multiplying through gives x²dy/dx + 2xy = x⁴, that is d/dx(yx²) = x⁴. Integrating, yx² = x⁵/5 + C, so y = x³/5 + C/x².
Marks-losing traps
Reporting a degree for an equation containing sin(dy/dx) or e^(dy/dx); mixing up order and degree; omitting C or introducing it twice; dividing by y or x without noticing the lost singular solution; applying the integrating-factor formula before the coefficient of dy/dx is made 1.