Internal problem ID [8712]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 376.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
\[ \boxed {{y^{\prime }}^{2}+a y^{\prime }+b y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 275
dsolve(diff(y(x),x)^2+a*diff(y(x),x)+b*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {a^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {c_{1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {c_{1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )}{4 b} y \left (x \right ) = -\frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {c_{1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {c_{1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )}{4 b} y \left (x \right ) = -\frac {{\mathrm e}^{-\frac {2 a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {c_{1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a \sqrt {-\frac {1}{b}}}\right )+a \ln \left (-\frac {1}{4 b}\right )-2 c_{1} b +2 x b +2 a}{2 a}} \left ({\mathrm e}^{-\frac {2 a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {c_{1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a \sqrt {-\frac {1}{b}}}\right )+a \ln \left (-\frac {1}{4 b}\right )-2 c_{1} b +2 x b +2 a}{2 a}}+2 a \right )}{4 b} \end{align*}
✓ Solution by Mathematica
Time used: 0.809 (sec). Leaf size: 119
DSolve[b*y[x] + a*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}+a \log \left (b \left (a-\sqrt {a^2-4 \text {$\#$1} b}\right )\right )}{2 b}\&\right ]\left [\frac {x}{2}+c_1\right ] y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {$\#$1} b}-a \log \left (b \left (\sqrt {a^2-4 \text {$\#$1} b}+a\right )\right )}{2 b}\&\right ]\left [-\frac {x}{2}+c_1\right ] y(x)\to 0 \end{align*}