Question

Make and solve the given equation x dx + y dy=a^{2}frac{x dy - y dx}{x^{2} + y^{2}}

Second order linear equations
ANSWERED
asked 2021-02-26
Make and solve the given equation \(x\ dx\ +\ y\ dy=a^{2}\frac{x\ dy\ -\ y\ dx}{x^{2}\ +\ y^{2}}\)

Answers (1)

2021-02-27
Given:\(x\ dx\ +\ y\ dy=a^{2}\frac{x\ dy\ -\ y\ dx}{x^{2}\ +\ y^{2}}\)
\(x\ dx\ +\ y\ dy=a^{2}\left(\frac{x\ dy\ -\ y\ dx}{x^{2}\ +\ y^{2}}\right)\)
\(=a^{2}\left[\frac{\frac{x\ dy\ -\ y\ dx}{x^{2}}}{\frac{x^{2}\ +\ y^{2}}{x^{2}}}\right]\)
\(=a^{2}\left[\frac{d\left(\frac{y}{x}\right)}{1\ +\ \frac{y^{2}}{x^{2}}}\right]\)
\(=a^{2}\left[\frac{d\left(\frac{y}{x}\right)}{1\ +\ \left(\frac{y}{x}\right)^{2}}\right]\)
\(\Rightarrow\ \int\ xdx\ +\ \int\ ydy=a^{2}\ \int\ \left[\frac{d\left(\frac{y}{x}\right)}{1\ +\ \left(\frac{y}{x}\right)^{2}}\right]\ +\ C\) where C is a constant
\(\Rightarrow\ \frac{x^{2}}{2}\ +\ \frac{y^{2}}{2}=a^{2}\ \tan^{-1}\left(\frac{y}{x}\right)\ C\ \left[\sin\ ce\ \int\ \frac{dx}{x^{2}\ +\ a^{2}}=\frac{1}{a^{2}}\ \frac{\tan^{-1}x}{a}\ +\ c\right]\)
Hence the solution of the given differential equation is \(x^{2}\ +\ y^{2}=2a^{2}\tan^{-1}\left(\frac{y}{x}\right)\)
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