Home page for accesible maths 6 Chapter 6 contents 6.18 First-order linear equations and integrating factors 6.20 General method continued

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

6.19 General method for first-order linear equations

The general method for solving a linear first-order equation is to multiply the whole equation by the integrating factor.

Method.

Integrating factor does the trick We claim that after multiplying by $I(x)$ , the left-hand side of the equation becomes $\frac{d}{dx}\left(I(x)y\right)$ .

To prove this, let $F(x)=\int p(x)\,dx$ , so that $I(x)=e^{F(x)}$ . Then $F^{\prime}(x)={p(x)}$ and, by the chain rule, $I^{\prime}(x)={F^{\prime}(x)e^{F(x)}=p(x)I(x).}$ Thus, by the product rule:

\frac{d}{dx}(I(x)y)={I(x)\frac{dy}{dx}+p(x)I(x)y=}{I(x)\left(\frac{dy}{dx}+p(x% )y\right).}

Caution: For this to work, the equation has to be in the form $\frac{dy}{dx}+p(x)y=q(x)$ , i.e. the coefficient of $dy/dx$ has to equal 1 (see 6.22 for what to do otherwise).