How to Solve Exact Second Order Differential Equations

Exact second order differential equations are solved in the same way as exact first order differential equations. You integrate the right and the left-hand side of the equation on their own. The only difference is that you must do it twice for the second order differential equation, as it contains second order derivatives.

\begin{array}{l} y^{″} & = f (x) \\ ⇓ \\ y^{'} & = \int f (x) d x \\ = F_{0} (x) + C_{1} \\ ⇓ \\ y & = \int y^{'} d x \\ = \int (F_{0} (x) + C_{1}) d x \\ = F (x) + C_{1} x + C_{2} \end{array}

\begin{array}{l} y^{″} & = f (x) \\ ⇓ \\ y^{'} & = \int f (x) d x = F_{0} (x) + C_{1} \\ ⇓ \\ y & = \int y^{'} d x = \int (F_{0} (x) + C_{1}) d x = F (x) + C_{1} x + C_{2} \end{array}

Example 1

Solve the differential equation $y^{″} = 20 x^{3} + e^{2 x}$

\begin{array}{l} y^{'} & = \int y^{″} d x \\ = \int 20 x^{3} + e^{2 x} d x \\ = 5 x^{4} + \frac{1}{2} e^{2 x} + C_{1} \\ y & = \int y^{'} d x \\ = \int 5 x^{4} + \frac{1}{2} e^{2 x} + C_{1} d x \\ = x^{5} + \frac{1}{4} e^{2 x} + C_{1} x + C_{2} \end{array}

\begin{array}{l} y^{'} & = \int y^{″} d x = \int 20 x^{3} + e^{2 x} d x = 5 x^{4} + \frac{1}{2} e^{2 x} + C_{1} \\ y & = \int y^{'} d x = \int 5 x^{4} + \frac{1}{2} e^{2 x} + C_{1} d x = x^{5} + \frac{1}{4} e^{2 x} + C_{1} x + C_{2} \end{array}