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Differential Equations - Solving Differential Equations by Separation of Variables

Differential Equations - Solving Differential Equations by Separation of Variables

Find the general solutions of the following equations:

  1. \({\frac{\mathrm{d}y}{\mathrm{d}x}={{e}^{3x+2y}}}\)
  2. \({\left( 4y+{{x}^{2}}y \right)\frac{\mathrm{d}y}{\mathrm{d}x}=\left( 2x+x{{y}^{2}} \right)}\)
  3. \({\frac{\mathrm{d}y}{\mathrm{d}x}=1+x+y+xy}\)
  4. \({\frac{\mathrm{d}y}{\mathrm{d}x}=x\sqrt{1-{{y}^{2}}}}\)

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Answer:(a) \({y=-\frac{1}{2}\ln\left\vert{}D-\frac{2}{3}\mathrm{e}^{3x}\right\vert{}}\) (b) \({y^2=k\left(x^2+4\right)-2}\) (c) \({y=A\mathrm{e}^{x+\frac{1}{2}x^2}-1}\) (d) \({\sin^{-1}y=\frac{x^2}{2}+c}\)

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