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Equation relating a function and its derivatives
D.E. with derivatives w.r.t. a single variable
D.E. with partial derivatives w.r.t. multiple variables
dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y)dxdy=g(x)h(y)
Separate variables and integrate
dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)dxdy+P(x)y=Q(x)
Use an integrating factor F(x)F(x)F(x)
F(x)=e∫P(x) dxF(x) = e^{\int P(x) \, dx}F(x)=e∫P(x)dx
M(x,y) dx+N(x,y) dy=0M(x, y) \, dx + N(x, y) \, dy = 0M(x,y)dx+N(x,y)dy=0
∂M∂y=∂N∂x\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}∂y∂M=∂x∂N
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