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- Exact Differential Equation
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Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$
Learn how to solve differential calculus problems step by step online.
$pdx1\left(x-pe^p\right)dp=0$
Learn how to solve differential calculus problems step by step online. Solve the differential equation dx/dp=(-(x-pe^p))/p. Rewrite the differential equation in the standard form M(x,y)dx+N(x,y)dy=0. The differential equation pdx1\left(x-pe^p\right)dp=0 is exact, since it is written in the standard form M(x,y)dx+N(x,y)dy=0, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and they satisfy the test for exactness: \displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form f(x,y)=C. Using the test for exactness, we check that the differential equation is exact. Integrate M(p,x) with respect to p to get.
