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Particular Solution of a Differential Equation

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Question

Solve : $\int \frac{x d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}$

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Solution

$\int \frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})} d x$

substitute $u = \frac{x^{2}}{2} \to d x = \frac{1}{x} d u$

$= \int \frac{1}{(2 u + a^{2}) (2 u + b^{2})} d u$

Upon partial fraction decomposition

$= \int (\frac{1}{(b^{2} - a^{2}) (2 u + a^{2})} - \frac{1}{(b^{2} - a^{2}) (2 u + b^{2})}) d u$

upon linearity

$= \frac{1}{(b^{2} - a^{2})} \int \frac{1}{2 u + a^{2}} d u - \frac{1}{(b^{2} - a^{2})} \int \frac{1}{2 u + b^{2}} d u$

$= \frac{1}{(b^{2} - a^{2})} \frac{ln (2 u + a^{2})}{2} - \frac{1}{(b^{2} - a^{2})} \frac{ln (2 u + b^{2})}{2}$

$= - \frac{ln (x^{2} + b^{2}) - ln (x^{2} + a^{2})}{2 (b^{2} - a^{2})} + C$

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\int \frac{x d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}

\int \frac{1}{(x^{2} + a^{2}) (x^{2} + b^{2})} d x

\int_{0}^{\infty} \frac{1}{(x^{2} + a^{2}) (x^{2} + b^{2})} d x

Q. Solve the following quadratic equation using quadratic formula:

a^{2} x^{2} + (a^{2} - b^{2}) x - b^{2} = 0

\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b^{2})} = k [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}] \Rightarrow k =

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