Linear differential equations

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Equations of the form

${a_n}(x){y^{(n)}} + .... + {a_2}(x)y'' + {a_1}(x)y' + {a_0}(x)y + b(x) = 0$

where y, the a´s and the b are functions of x, and n>0 are called linear differential equations. The term linear comes from that the whole expression is a linear combination of derivatives, for a given x at least.

We will here look at some special cases. To start with, what can we solve more or less directly? Say we have

$f(x){y^{(n)}} + g(x) = 0$

This can be directly rewritten as

${y^{(n)}} = - \frac{{g(x)}}{{f(x)}}$

that can then be solved directly by n integrations (if possible).

So if we have

$f(x)y' + g(x)y = 0$

then we can separate the variables to get

$\frac{{dy}}{y} = - \frac{{g(x)}}{{f(x)}}dx$

that also can then be solved directly by integration (if possible).

In the next few pages, we will look at a few more special cases.

Up a level : Differential Equations
Previous page : Homogenous differential equations – a first type
Next page : First order linear differential equations

Last modified: Dec 28, 2023 @ 16:44