# Linear Second Order ODE/y'' = y'/Proof 1

## Theorem

The second order ODE:

$(1): \quad y'' = y'$

has the general solution:

$y = A_1 e^x + A_2$

## Proof

The proof proceeds by using Solution of Second Order Differential Equation with Missing Dependent Variable.

Substitute $p$ for $y'$ in $(1)$:

 $\ds \dfrac {\d p} {\d x}$ $=$ $\ds p$ where $p = \dfrac {\d y} {\d x}$ $\ds \leadsto \ \$ $\ds \int \rd x$ $=$ $\ds \int \frac {\d p} p$ Separation of Variables $\ds \leadsto \ \$ $\ds x$ $=$ $\ds \ln p + C$ Primitive of $\dfrac 1 x$ $\ds \leadsto \ \$ $\ds p = \frac {\d y} {\d x}$ $=$ $\ds A_1 e^x$ where $A_1 = e^C$ $\ds \leadsto \ \$ $\ds \int \rd y$ $=$ $\ds \int A_1 e^x \rd x$ Separation of Variables $\ds \leadsto \ \$ $\ds y$ $=$ $\ds A_1 e^x + A_2$ Primitive of Exponential Function

$\blacksquare$