Differential Equations
Questions here ask for the order and degree of an equation, formation of an equation from a solution, and solution by the variable separable method or by an integrating factor. Applied parts link the equation to a growth, decay or cost situation.
Order, degree and solutions
Order is the order of the highest derivative present. Degree is the power of the highest derivative once the equation is cleared of radicals and fractions in derivatives. For example, (d²y/dx²)³ + (dy/dx)⁴ = 0 has order 2 and degree 3. A general solution contains as many arbitrary constants as the order; a particular solution uses the given condition to fix them.
Variable separable
Rewrite dy/dx = f(x)g(y) and integrate: ∫dy/g(y) = ∫f(x)dx. Apply the initial condition only after integrating.
First-order linear
For dy/dx + Py = Q, with P and Q functions of x, the integrating factor is e^{∫P dx} and the solution is y(IF) = ∫Q(IF)dx + c.
Worked example
Solve dy/dx = (x + 1)/y with y = 2 when x = 0. Separate: y dy = (x + 1)dx. Integrate: y²/2 = x²/2 + x + c. Put x = 0, y = 2: 4/2 = 0 + 0 + c, so c = 2. Multiplying by 2 gives y² = x² + 2x + 4. Check at x = 0: y² = 4, and y = 2 (positive root taken because y(0) = 2).
Traps
- Stating degree before removing a radical or fractional power of a derivative.
- Applying the initial condition before integrating, or losing the constant c.
- Mis-separating, for example treating (x + 1)/y as (x + 1)y.
- Choosing the wrong sign of the square root when the given condition is positive.