How to Solve EXACT Differential Equations

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Definition of Exact Equation • A differential equation of type • Mdx+Ndy=0 where M and N are all functions of x • is called an exact differential equation if there exists a function of two variables F(x,y) with continuous partial derivatives such that • dF(x,y) = Mdx+Ndy. • The general solution of an exact equation is given by • F(x,y)=C, • where C is an arbitrary constant. • Test for Exactness • Let functions M and N have continuous partial derivatives in a certain domain D. The differential equation Mdx+Ndy =0 is an exact equation if and only if • ∂M/∂y=∂N/∂x. • Algorithm for Solving an Exact Differential Equation • First it’s necessary to make sure that the differential equation is exact using the test for exactness: • ∂M/∂y=∂N/∂x. • Then we write the system of two differential equations that define the function F(x,y): • ∂F/∂x=M and ∂F/∂y=N • Integrate the first equation over the variable x. Instead of the constant C, we write an unknown function of y: • F(x,y) = ∫M(x,y)dx. • The we differentiating with respect to y, we substitute the function F(x,y)into the second equation: • ∂F/∂y = ∂/∂y[∫M(x,y)dx+φ(y)]=N(x,y). • From here we get expression for the derivative of the unknown function φ(y): • φ′(y)=M−∂F/∂y • By integrating the last expression, we find the function φ(y) and, hence, the function F(x,y): • F(x,y) = ∫Mdx+φ(y). • The general solution of the exact differential equation is given by • 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