Question

Solve the ODE x(dy/dx)=y-(x^2+y^2)^1/2

Answer 1

dy/dx = (y/x) - √(1 + (y/x)²).

So set y = xu. Then dy/dx = x du/dx + u ==>

x du/dx + u = u - √(1 + u²) ==> du/√(1 + u²) = - dx/x.

You can use a trigonometric substitution to integrate the left side. Putting u = tanΘ gives

∫ du/√(1 + u²) = ∫ secΘ dΘ = ln|secΘ + tanΘ| = ln|√(1 + u²) + u|.

ln|√(1 + u²) + u| = - ln|x| + C ==> ln|√(1 + u²) + u| + ln|x| = C

Use properties of logs (ln(a) + ln(b) = ln(ab)) and substitute u = y/x to get

ln|√(x² + y²) + y| = C ==> √(x² + y²) + y = A

where A = e^C.

Answer 2

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Answer 3

Thanks a lot.. :)