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Solve the differential equation: {eq}(e^{x}\sin y+3y)dx+(3x-\cos y+e^{x}\cos y)dy=0 {/eq}.

Question:

Solve the differential equation: {eq}(e^{x}\sin y+3y)dx+(3x-\cos y+e^{x}\cos y)dy=0 {/eq}.

Exact Equation:

A differential equation is exact if the functions that define it keep the relation:

{eq}P\left( {x,y} \right)dx + Q\left( {x,y} \right)dy = 0\\ \frac{{\partial P}}{{\partial y}}\left( {x,y} \right) = \frac{{\partial Q}}{{\partial x}}\left( {x,y} \right) {/eq}

The solution is obtained from a function that is called potential function, which meets the following relationship:

{eq}f\left( {x,y} \right) \to \left\{ \begin{array}{l} \frac{{\partial f}}{{\partial x}}\left( {x,y} \right) = P\left( {x,y} \right)\\ \frac{{\partial f}}{{\partial y}}\left( {x,y} \right) = Q\left( {x,y} \right) \end{array} \right. {/eq}

Answer and Explanation: 1

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First we check that the equation is exact:

{eq}\left( {{e^x}\sin y + 3y} \right)dx + \left( {3x - \cos y + {e^x}\cos y} \right)dy = 0\\ \left\{...

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