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

$\blacksquare$

Sources

• 1952: H.T.H. Piaggio: An Elementary Treatise on Differential Equations and their Applications (revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: Examples for solution: $(3)$