The dalembertsol routine determines whether the first argument is a first order ODE of d'Alembert type and, if so, returns a solution to the equation.

The first argument is a differential equation in diff or D form. The second argument (required cannot be inferred from the differential equation) is the variable in the differential equation.

This function is part of the DEtools package, and so it can be used in the form dalembertsol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[dalembertsol](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{dAlembert\_ode}≔y\left(x\right)=xf\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\right)+g\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\right)$

$\mathrm{dAlembert\_ode}≔y\left(x\right)=xf\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+g\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{dalembertsol}\left(\mathrm{dAlembert\_ode}\,y\left(x\right)\right)$

$\left\{y\left(x\right)=x\mathrm{RootOf}\left(-f\left(\mathrm{\_Z}\right)+\mathrm{\_Z}\right)+g\left(\mathrm{RootOf}\left(-f\left(\mathrm{\_Z}\right)+\mathrm{\_Z}\right)\right)\,\left[x\left(\mathrm{\_T}\right)={\ⅇ}^{\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}}\left(\int \frac{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_T}}g\left(\mathrm{\_T}\right)\right){\ⅇ}^{-\left(\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}\right)}}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}\right)\,y\left(x\right)\left(\mathrm{\_T}\right)={\ⅇ}^{\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}}\left(\int \frac{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_T}}g\left(\mathrm{\_T}\right)\right){\ⅇ}^{-\left(\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}\right)}}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}\right)f\left(\mathrm{\_T}\right)+g\left(\mathrm{\_T}\right)\right]\right\}$