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प्रश्न
Solve the following differential equation:
`x dy/dx = y - x tan (y/x)`
योग
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उत्तर
`x dy/dx = y - x tan (y/x)`
`dy/dx = y/x - tan (y/x)`
Let y = vx, where v is a function of x.
⇒ `dy/dx = v + x (dv)/dx`
Substitute in the differential equation:
⇒ `v + x (dv)/dx = v - tanv`
⇒ `x(dv)/dx = - tan v`
⇒ `(dv)/tanv = - (dx)/x`
Integrating both sides
⇒ `int cot v dv = - int dx/x`
⇒ `log|sin v| = -logx + logc`
⇒ `log|sin y/x| = -logx + logc`
⇒ `log|sin y/x| = log c/x`
⇒ `|sin y/x| = c/x`
⇒ `x sin y/x = c`
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