Internal problem ID [3511]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 27
Problem number: 779.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+a y^{\prime }+b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.141 (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 \relax (x ) = -\frac {a^{2} \left (\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 ) \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 \relax (x ) = -\frac {a^{2} \left (\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 ) \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 \relax (x ) = -\frac {{\mathrm e}^{-\frac {2 a \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 b x +2 a}{2 a}} \left ({\mathrm e}^{-\frac {2 a \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 b x +2 a}{2 a}}+2 a \right )}{4 b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.782 (sec). Leaf size: 119
DSolve[(y'[x])^2+a y'[x]+b y[x]==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*}