Question

Determine whether or not the following equation is exact, find the solution. If not, use an integrating factor to convert it into exact and solve it. 2e^(x/2 +y) - y + 2((xe ^(x/2 +y) + 1) )dy/dx = 0 , y(0) = 1

Answer 1

A DE is exact iff, when in the form \[M dx + Ndy = 0\]that \(\frac{\partial M}{\partial x} = \frac{\partial N}{\partial x}\) because by Young's theorem this implies the existence of a relation \(\Psi(x, y) = C\) which can in turn be solved for \(y\).

Answer 2

\[2e^{x/2 +y} - y + 2(xe ^{x/2 +y} + 1) \frac{\mathrm dy}{\mathrm dx} = 0\]

\[(2e^{x/2 +y} - y )\mathrm dx+ 2(xe ^{x/2 +y} + 1) \mathrm dy= 0\]