Differential Equation

Steps to Solve the Exact Differential Equation M dx + N dy = 0

1. Check the exactness condition:

\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

If satisfied, the equation is exact.

2. Find the potential function F(x, y) :

· Integrate M with respect to x :

F(x, y) = \int M(x, y) \, dx + g( y )

· Differentiate the result with respect to y and equate it to N :

\frac{\partial F}{\partial y} = \frac{\partial}{\partial y} \left[ \int M \, dx \right] + g'( y) = N(x, y)

· Solve for g'( y) :

g'( y) = N(x, y) - \frac{\partial}{\partial y} \left[ \int M \, dx \right]

· Integrate g'( y) to obtain g( y) :

g(y ) = \int g'( y) \dy

3. Substitute g(y ) into F(x, y) :

F(x, y) = \int M(x, y) \, dx + \int \left[ N(x, y) - \frac{\partial}{\partial y} \left( \int M \, dx \right) \right] dy

4. Write the general solution:

F(x, y) = C

where C is constant.

Do exact differential equations belong to PDEs or ODEs?