# 2.3 | Linear Equations

Method for Solving Linear Equations

1. Write the equation in standard form
$$\frac{dy}{dx} + P(x) y = Q(x).$$
2. Calculate the integrating factor $$\mu(x)$$ by the formula
$$\mu(x) = e^{\int P(x) \; dx}.$$
3. Multiply the equation in standard form by $$\mu(x)$$ and, recalling that the left-hand side is just $$\frac{d}{dx}[\mu(x) y ],$$ obtain
$$\begin{array}{c c c} \underbrace{\mu(x) \frac{dy}{dx} + P(x) \mu(x) y} & = & \mu(x) Q(x), \\ \frac{d}{dx}\left[ \mu(x) y \right] & = & \mu(x) Q(x). \end{array}$$
4. Integrate the last equation and solve for $$y$$ by dividing by $$\mu(x)$$ to obtain
$$y(x) = \frac{1}{\mu(x)} \left[ \int \mu(x) Q(x) \; dx + C \right].$$
Existence and Uniqueness of Solution
Suppose $$P(x)$$ and $$Q(x)$$ are continuous on an interval $$(a,b)$$ that contains the point $$x_0$$. Then for any choice of initial values $$y_0,$$ there exists a unique solution $$y(x)$$ on $$(a,b)$$ to the initial value problem
$$\frac{dy}{dx} + P(x) y = Q(x), \quad y(x_0) = y_0.$$
In fact, the solution is given by
$$y(x) = \frac{1}{\mu(x)} \left[ \int \mu(x) Q(x) \; dx + C \right],$$
for some constant $$C$$.