How to Find the General Solution of Differential Equation
Problems with differential equations are asking you to find an unknown function or functions, rather than a number or set of numbers as you would normally find with an equation like f(x) = x2 + 9.
For example, the differential equation dy⁄dx = 10x is asking you to find the derivative of some unknown function y that is equal to 10x.
General Solution of Differential Equation: Example
Example problem #1: Find the general solution for the differential equation dy⁄dx = 2x.
Step 1: Use algebra to get the equation into a more familiar form for integration:
dy⁄dx = 2x →
dy = 2x dx
Step 2: Integrate both sides of the equation:
∫dy = ∫2x dx →
&int1 dy = &int2x dx →
y = x2 + C
Example problem #2: Find the general solution for the differential equation dy⁄dx = x2 – 3
Step 1: Use algebra to get the equation into a more familiar form for integration:
dy⁄dx = x2 – 3→
dy = x2 – 3 dx
Step 2: Integrate both sides of the equation:
∫ dy = ∫x2 – 3 dx →
∫ 1 dy = ∫x2 – 3 dx →
y = x3⁄3 -3x + C
Sample problem #3: Find the general solution for the differential equation θ2 dθ = sin(t + 0.2) dt.
Step 1: Integrate both sides of the equation:
∫ θ2 dθ = ∫sin(t + 0.2) dt →
θ3 = -cos(t + 0.2) + C
That’s how to find the general solution of differential equations!
Tip: If your differential equation has a constraint, then what you need to find is a particular solution. For example, dy⁄dx = 2x ; y(0) = 3 is an initial value problem that requires you to find a solution that satisfies the constraint y(0) = 3.
References
Larson & Edwards. Calculus.