Formation of Ordinary Differential Equation by Elimination/Examples/y equals A e^Bx

Examples of Formation of Ordinary Differential Equation by Elimination

Consider the equation:

$(1): \quad y = A e^{B x}$

This can be expressed as the ordinary differential equation of order $2$:

$\dfrac {\d^2 y} {\d x^2} + 9 y = 0$

Proof

Differentiating twice with respect to $x$:

 $\ds \dfrac {\d y} {\d x}$ $=$ $\ds B A e^{B x}$ Derivative of Exponential Function $\text {(2)}: \quad$ $\ds$ $=$ $\ds B y$ Derivative of Exponential Function $\ds \leadsto \ \$ $\ds \dfrac {\d^2 y} {\d x^2}$ $=$ $\ds B \dfrac {\d y} {\d x}$ $\ds \leadsto \ \$ $\ds \dfrac {\d^2 y} {\d x^2}$ $=$ $\ds \dfrac {\paren {\frac {\d y} {\d x} }^2} y$ substituting for $B$ from $2$ $\ds \leadsto \ \$ $\ds y \cdot \dfrac {\d^2 y} {\d x^2}$ $=$ $\ds \paren {\frac {\d y} {\d x} }^2$ simplifying

$\blacksquare$