In class, we defined a separable differential equation to be one of the form dy/

Question

In class, we defined a separable differential equation to be one of the form dy/dx = g(x) h(y) for some functions g(x) and h(y). This equation can be rearranged to look like -g(x) + y'/h(y) = 0. Show that this equation is exact. Hence every separable equation is exact. For the exact equation in the previous part, what integrals would need to be computed to determined the function si(x,y)? (Express the integrals in terms of the functions g(x) and h(y).) Let's return to the original separable equation dy/dx = g(x) h(y). Using our method for separable equations, what integrals would need to be computed to solve this ODE? (Compare with the previous part.)

Explanation / Answer

The above can be rearranged as -g(x)dx+dy/h(y)=0

compare the above equation with Mdx+Ndy=0

here M=-g(x),N=1/h(y)

then My=Nx=0

therefore given differential equation is exact

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