Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0/Proof 1
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Theorem
The second order ODE:
- $(1): \quad \paren {x^2 + x} y'' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$
has the general solution:
- $y = C_1 e^x + \dfrac {C_2} x$
Proof
Note that:
\(\ds y_1\) | \(=\) | \(\ds \frac 1 x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds {y_1}'\) | \(=\) | \(\ds -\frac 1 {x^2}\) | Power Rule for Derivatives | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds {y_1}''\) | \(=\) | \(\ds \frac 2 {x^3}\) | Power Rule for Derivatives |
and so:
\(\ds \) | \(\) | \(\ds \paren {x^2 + x} {y_1}'' + \paren {2 - x^2} {y_1}' - \paren {2 + x} y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \frac {x^2 + x} {x^3} - \frac {2 - x^2} {x^2} - \frac {2 + x} x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 x + \frac 2 {x^2} - \frac 2 {x^2} + 1 - \frac 2 x - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Thus:
- $y_1 = \dfrac 1 x$
is a particular solution of $(1)$.
$(1)$ can be expressed as:
- $(2): \quad y'' + \dfrac {2 - x^2} {x^2 + x} y' - \dfrac {2 + x} {x^2 + x} y = 0$
which is in the form:
- $y'' + \map P x y' + \map Q x y = 0$
where:
- $\map P x = \dfrac {2 - x^2} {x^2 + x}$
- $\map Q x = \dfrac {2 + x} {x^2 + x}$
From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another:
- $\map {y_2} x = \map v x \, \map {y_1} x$
where:
- $\ds v = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x$
is also a particular solution of $(1)$.
We have that:
\(\ds \int P \rd x\) | \(=\) | \(\ds \int \paren {\dfrac {2 - x^2} {x^2 + x} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \dfrac 2 {x \paren {x + 1} } \rd x - \int \paren {\dfrac x {x + 1} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \, \map \ln {\frac x {x + 1} } - \int \paren {\dfrac x {x + 1} } \rd x\) | Primitive of $\dfrac 1 {x \paren {a x + b} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \, \map \ln {\frac x {x + 1} } - x + \, \map \ln {x + 1}\) | Primitive of $\dfrac x {a x + b}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\frac {x^2} {x + 1} } - x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\frac {e^{-x} x^2} {x + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\map \ln {\frac {e^x \paren {x + 1} } {x^2} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds e^{-\int P \rd x}\) | \(=\) | \(\ds e^{\map \ln {\frac {e^x \paren {x + 1} } {x^2} } }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^x \paren {x + 1} } {x^2}\) |
Hence:
\(\ds v\) | \(=\) | \(\ds \int \dfrac 1 { {y_1}^2} e^{- \int P \rd x} \rd x\) | Definition of $v$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int x^2 \paren {\frac {e^x \paren {x + 1} } {x^2} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int e^x \paren {x + 1} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int x e^x \rd x + \int e^x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^x \paren {x - 1} + \int e^x \rd x\) | Primitive of $x e^{a x}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds e^x \paren {x - 1} + e^x\) | Primitive of $e^{a x}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x e^x\) |
and so:
\(\ds y_2\) | \(=\) | \(\ds v y_1\) | Definition of $y_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x e^x} \frac 1 x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^x\) |
From Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution:
- $y = C_1 e^x + \dfrac {C_2} x$
$\blacksquare$