2.3 Exact Equations and Integrating Factors

Exact equations and integrating factors are powerful tools for solving first-order differential equations. They build on the concept of separable equations, offering methods to tackle more complex problems that can't be solved by simple separation of variables.

These techniques expand our problem-solving toolkit, allowing us to solve a wider range of differential equations. By recognizing exact equations and using integrating factors, we can transform tricky equations into solvable forms, making them easier to handle.

Exact Equations

Definition and Characteristics

• Exact differential equation takes the form $M(x,y)dx + N(x,y)dy = 0$
  • $M(x,y)$ represents the coefficient of $dx$
  • $N(x,y)$ represents the coefficient of $dy$
• Exactness condition states that if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, then the equation is exact
  • Partial derivatives of $M$ with respect to $y$ and $N$ with respect to $x$ must be equal for the equation to be exact
• Mixed partial derivatives are used to verify the exactness condition
  • Second-order mixed partial derivatives $\frac{\partial^2 f}{\partial x \partial y}$ and $\frac{\partial^2 f}{\partial y \partial x}$ are equal for a function $f(x,y)$ with continuous second-order partial derivatives

Solution Methods

• Potential function $\phi(x,y)$ exists for an exact differential equation
  • $\phi(x,y)$ satisfies $\frac{\partial \phi}{\partial x} = M(x,y)$ and $\frac{\partial \phi}{\partial y} = N(x,y)$
• To find the potential function, integrate $M(x,y)$ with respect to $x$, treating $y$ as a constant
  • $\phi(x,y) = \int M(x,y) dx + g(y)$, where $g(y)$ is an arbitrary function of $y$
• Differentiate the potential function with respect to $y$ and equate it to $N(x,y)$ to determine $g(y)$
  • $\frac{\partial \phi}{\partial y} = N(x,y)$ gives an equation to solve for $g(y)$
• The general solution of the exact differential equation is given by $\phi(x,y) = C$, where $C$ is an arbitrary constant

Integrating Factors

Definition and Purpose

• Integrating factor is a function $\mu(x,y)$ that, when multiplied by a non-exact differential equation, makes it exact
  • Multiplying a non-exact equation by an appropriate integrating factor transforms it into an exact equation
• The existence of an integrating factor allows the solution of non-exact differential equations
  • Many non-exact equations can be solved by finding a suitable integrating factor

Finding Integrating Factors

• For a linear first-order differential equation $\frac{dy}{dx} + P(x)y = Q(x)$, the integrating factor is $\mu(x) = e^{\int P(x) dx}$
  • Multiplying the equation by $\mu(x)$ makes it exact
• For a homogeneous first-order differential equation $\frac{dy}{dx} = f(\frac{y}{x})$, the integrating factor is $\mu(x,y) = \frac{1}{xy}$
  • Multiplying the equation by $\mu(x,y)$ transforms it into an exact equation
• In some cases, the integrating factor may depend on both $x$ and $y$, such as $\mu(x,y) = x^ny^m$
  • The values of $n$ and $m$ can be determined by solving a system of equations derived from the exactness condition

Solution Process

• After finding the appropriate integrating factor, multiply the non-exact differential equation by it
  • The resulting equation becomes exact
• Apply the solution methods for exact differential equations to find the general solution
  • Find the potential function and set it equal to an arbitrary constant
• The implicit solution is obtained by solving the equation $\phi(x,y) = C$ for $y$ in terms of $x$
  • Explicit solutions may not always be possible, depending on the complexity of the equation