4.5 First-order linear equations  (Page 2/10)

Examples of first-order nonlinear differential equations include

These equations are nonlinear because of terms like ${\left(y^{\prime }\right)}^4,y^3,$ etc. Due to these terms, it is impossible to put these equations into the same form as [link] .

Standard form

Consider the differential equation

Our main goal in this section is to derive a solution method for equations of this form. It is useful to have the coefficient of $y^{\prime }$ be equal to $1.$ To make this happen, we divide both sides by $3x^2-4.$

This is called the standard form    of the differential equation. We will use it later when finding the solution to a general first-order linear differential equation. Returning to [link] , we can divide both sides of the equation by $a\left(x\right).$ This leads to the equation

Now define $p\left(x\right)=\frac{b\left(x\right)}{a\left(x\right)}$ and $q\left(x\right)=\frac{c\left(x\right)}{a\left(x\right)}.$ Then [link] becomes

We can write any first-order linear differential equation in this form, and this is referred to as the standard form for a first-order linear differential equation.

Integrating factors

We now develop a solution technique for any first-order linear differential equation. We start with the standard form of a first-order linear differential equation:

The first term on the left-hand side of [link] is the derivative of the unknown function, and the second term is the product of a known function with the unknown function. This is somewhat reminiscent of the power rule from the Differentiation Rules section. If we multiply [link] by a yet-to-be-determined function $\mu \left(x\right),$ then the equation becomes

The left-hand side [link] can be matched perfectly to the product rule:

Matching term by term gives $y=f\left(x\right),g\left(x\right)=\mu \left(x\right),$ and $g^{\prime }\left(x\right)=\mu \left(x\right)p\left(x\right).$ Taking the derivative of $g\left(x\right)=\mu \left(x\right)$ and setting it equal to the right-hand side of $g^{\prime }\left(x\right)=\mu \left(x\right)p\left(x\right)$ leads to

This is a first-order, separable differential equation for $\mu \left(x\right).$ We know $p(x)$ because it appears in the differential equation we are solving. Separating variables and integrating yields