First Order Differential Equations

Linear Equations

A first order differential equation is one that can be put forward in the form

$$\frac{dy}{dx}+P(x)y=Q(x)$$

Solving with $v(x)$

Into this equation we introduce a new function $v(x)$ such that $v\frac{dy}{dx}+Pvy=\frac{d}{dx}(v\cdot y)$ and the equation becomes

$$\frac{d}{dx}(v(x)\cdot y)=v(x)Q(x)$$

Integrating both sides and dividing by $v(x)$ yields

$$y=\frac{1}{v(x)}\int v(x)Q(x)dx$$

expressing the solution in terms of the function $v(x)$ and $Q(x)$. $v(x)$ is the integrating factor for the equation.