Second Order ODE/y'' = f(x)

Theorem

The second order ODE:

$\dfrac {\d^2 y} {\d x^2} = \map f x$

has the general solution:

$\ds y = \iint \map f x \rd x \rd x + C_1 x + C_2$

where $C_1$ and $C_2$ are arbitrary constants.

Proof

\(\ds \dfrac {\d^2 y} {\d x^2}\)

\(=\)

\(\ds \map f x\)

\(\ds \leadsto \ \ \)

\(\ds \int \dfrac {\d^2 y} {\d x^2} \rd y\)

\(=\)

\(\ds \int \map f x \rd x\)

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \int \map f x \rd x + C_1\)

where $C_1$ is an arbitrary constant

\(\ds \leadsto \ \ \)

\(\ds \int \dfrac {\d y} {\d x} \rd y\)

\(=\)

\(\ds \int \paren {\int \map f x \rd x + C_1} \rd x\)

Solution to Separable Differential Equation

\(\ds \)

\(=\)

\(\ds \int \paren {\int \map f x \rd x} \rd x + \int C_1 \rd x\)

Linear Combination of Primitives

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds \iint \map f x \rd x \rd x + C_1 x + C_2\)

Primitive of Power: $C_2$ is another arbitrary constant

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation: differential equations of the second order: $(1)$ Equations of the form $\dfrac {\d^2 y} {\d x^2} = \map f x$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation: differential equations of the second order: $(1)$ Equations of the form $\dfrac {\d^2 y} {\d x^2} = \map f x$