The general form of the Lagerstrom ODE is given by the following:

Lagerstrom_ode := diff(y(x),x,x)= -k*diff(y(x),x)/x-epsilon*y(x)*diff(y(x),x);

$\mathrm{Lagerstrom\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=-\frac{k\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{x}-\mathrm{\epsilon }y\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

See Rosenblat and Shepherd, "On the Asymptotic Solution of the Lagerstrom Model Equation".

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

$\mathrm{odeadvisor}\left(\mathrm{Lagerstrom\_ode}\right)$

$\left[\mathrm{\_Lagerstrom}\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{symgen}\left(\mathrm{Lagerstrom\_ode}\,\mathrm{way}=3\right)$

$\left[\mathrm{\_\ξ}=-x\,\mathrm{\_\η}=y\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Lagerstrom\_ode}\,\mathrm{way}=3\right)$

$\mathrm{ans}≔y\left(x\right)=\left(\mathrm{\_a}{\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right)\mathbf{where}\left[\left\{\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=\left(-{\mathrm{\_a}}^2\mathrm{\epsilon }-\mathrm{\_a}k+2\mathrm{\_a}\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^3+\left(-\mathrm{\epsilon }\mathrm{\_a}-k+3\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^2\right\}\,\left\{\mathrm{\_a}=y\left(x\right)x\,\mathrm{\_b}\left(\mathrm{\_a}\right)=-\frac{1}{x\left(\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x+y\left(x\right)\right)}\right\}\,\left\{x=\frac{1}{{\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}}\,y\left(x\right)=\mathrm{\_a}{\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right\}\right]$

$\mathrm{odetest}\left(\mathrm{ans}\,\mathrm{Lagerstrom\_ode}\right)$

$0$

$\mathrm{reduced\_ode}≔\mathrm{op}\left(\left[2\,2\,1\,1\right]\,\mathrm{ans}\right)$

$\mathrm{reduced\_ode}≔\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=\left(-{\mathrm{\_a}}^2\mathrm{\epsilon }-\mathrm{\_a}k+2\mathrm{\_a}\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^3+\left(-\mathrm{\epsilon }\mathrm{\_a}-k+3\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^2$

$\mathrm{odeadvisor}\left(\mathrm{reduced\_ode}\right)$

$\left[\mathrm{\_Abel}\right]$