Solution to Differential Equation/Examples/Arbitrary Order 2 ODE
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Examples of Solutions to Differential Equations
Consider the real function defined as:
- $y = \map f x = \ln x + x$
defined on the domain $x \in \R_{>0}$.
Then $\map f x$ is a solution to the second order ODE:
- $(1): \quad x^2 y'' + 2 x y' + y = \ln x + 3 x + 1$
defined on the domain $x \in \R_{>0}$.
Proof
It is noted that $\map f x$ is not defined in $\R$ when $x \le 0$ because that is outside the domain of the natural logarithm $\ln$.
The same constraint applies to $(1)$.
Having established that, we continue:
\(\ds y\) | \(=\) | \(\ds \ln x + x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y'\) | \(=\) | \(\ds \dfrac 1 x + 1\) | Derivative of Natural Logarithm, Derivative of Identity Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y''\) | \(=\) | \(\ds -\dfrac 1 {x^2} + 0\) | Power Rule for Derivatives, Derivative of Constant |
Then:
\(\ds \) | \(\) | \(\ds x^2 \paren {-\dfrac 1 {x^2} } + 2 x \paren {\dfrac 1 x + 1} + \ln x + x\) | substituting for $y$, $y'$ and $y''$ from above into the left hand side of $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -1 + 2 + 2 x + \ln x + x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln x + 3 x + 1\) | which equals the right hand side of $(1)$ |
$\blacksquare$
Sources
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation