linear - Maple Help

Solving Linear ODEs

Description

 • The general form of a first order linear ODE is given by the following:
 > linear_ode := diff(y(x),x)+f(x)*y(x)-g(x);
 ${\mathrm{linear_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{f}{}\left({x}\right){}{y}{}\left({x}\right){-}{g}{}\left({x}\right)$ (1)
 where f(x) and g(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 16. This type of ODE can be solved in a general manner by dsolve as follows:

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{linear_ode}\right)$
 $\left[{\mathrm{_linear}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{linear_ode}\right)$
 ${y}{}\left({x}\right){=}\left({\int }{g}{}\left({x}\right){}{{ⅇ}}^{{\int }{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{\mathrm{_C1}}\right){}{{ⅇ}}^{{\int }{-}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (4)