## 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) = x^{2} + 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 = x^{2} + C

**Example problem #2: **Find the general solution for the differential equation ^{dy}⁄_{dx} = x^{2} – 3

Step 1: **Use algebra** to get the equation into a more familiar form for integration:

^{dy}⁄_{dx} = x^{2} – 3→

dy = x^{2} – 3 dx

Step 2: **Integrate both sides** of the equation:

∫ dy = ∫x^{2} – 3 dx →

∫ 1 dy = ∫x^{2} – 3 dx →

y = ^{x3}⁄_{3} -3x + C

**Sample problem #3: **Find the general solution for the differential equation θ^{2} dθ = sin(t + 0.2) dt.

Step 1: **Integrate both sides** of the equation:

∫ θ^{2} dθ = ∫sin(t + 0.2) dt →

θ^{3} = -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.