Practice Problems: Exact DEs

Exact differential equations are a special type of differential equation that can be solved using a unique technique known as the method of integrating factors. An exact differential equation can be identified by checking whether the partial derivatives of the equation with respect to its variables are equal. In this tutorial, we will explore the steps involved in solving exact differential equations, including identifying an equation as exact, finding the integrating factor, and solving for the general solution

The general formula for an exact differential equation is \(M(x,y)dx + N(x,y)dy = 0 \) where \(M(x,y)\) and \(N(x,y)\) are functions of \(x\)and \(y\). To solve an exact differential equation, we need to determine whether there exists a function \(\psi(x,y)\) such that \(\frac{\partial \psi}{\partial x} = M(x,y), \quad \frac{\partial \psi}{\partial y} = N(x,y)\)

If such a function exists, then the differential equation is exact, and we can solve it by finding \(\psi(x,y)\) and setting it equal to a constant \(C\). However, if such a function does not exist, we need to find an integrating factor \(\mu(x,y)\) that will transform the differential equation into an exact one. The integrating factor is given by:

\(\mu(x,y) = e^{\int P(x)dx}\) where \(P(x)\) is the coefficient of \(dx\) in the expression for \(M(x,y)dx + N(x,y)dy = 0\). By multiplying both sides of the differential equation by \(\mu(x,y)\) and applying the method of integrating factors, we can find \(\psi(x,y)\) and solve for the general solution of the differential equation.

Solving exact differential equations.

See Integral Formulas.