Complex numbers and ODEs: problem set 2 (basic solution techniques for ODEs)

3. Solving first order ODEs

(iv) Solve $xy y' - y^2 = (x + y)^2 e^{-y/x}$

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(viii) Solve $\frac{dx}{dy} = \cos(2y) - x\cot y$

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4. Solving a nonlinear ODE

We wish to solve \[ y' + ky = y^n \sin x. \]

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5. Solving an exact ODE

We wish to solve \[ \frac{dy}{dx} = \frac{(3x^2 + 2xy + y^2) \sin x + (6x + 2y)\cos x}{(2x + 2y)\cos x}. \]

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