Differential Equations
Differential Equations tests recognition of order and degree, formation of equations from families of curves, and solution of first-order equations, especially variable-separable and linear forms. The exam rewards correct classification before integration.
Core rules
- Order is the highest derivative present; degree is the power of that highest derivative after the equation is made polynomial in derivatives.
- A general solution contains as many arbitrary constants as the order; a particular solution fixes them from initial conditions.
- Variable separable: write dy/dx = f(x)g(y), separate as dy/g(y)=f(x)dx, then integrate both sides.
- Linear equation: dy/dx + P(x)y = Q(x). The integrating factor is IF = e^(∫P dx), and the solution is y·IF = ∫Q·IF dx + C.
- Formation: differentiate the given family enough times to eliminate all arbitrary constants.
Worked example
Solve dy/dx + y/x = x, x>0. Here P=1/x, so IF=e^(∫(1/x)dx)=e^(ln x)=x. Multiply: x dy/dx + y = x^2, that is d(xy)/dx=x^2. Integrate: xy=x^3/3+C, so y=x^2/3+C/x. Check by differentiating: y' = 2x/3 - C/x^2; then y'+y/x = 2x/3 - C/x^2 + x/3 + C/x^2 = x.
Mark-losing traps
- Calling an equation linear when y is squared or multiplied by y'.
- Using the wrong sign in the integrating factor.
- Forgetting the constant of integration until the final step.
- Losing a solution when dividing by g(y), for example y=0.
- Stating degree when the equation is not polynomial in derivatives.