Internal problem ID [4019]
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]
\[ \boxed {{y^{\prime }}^{2}+y^{\prime } a +y b=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 245
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 {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )}{4 b} \\ y \left (x \right ) &= -\frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a}\right )}{4 b} \\ y \left (x \right ) &= {\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-a +\left (c_{1} -x \right ) b}{a}} \left (a \sqrt {-\frac {1}{b}}+{\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {\left (c_{1} -x \right ) b -a}{a}}}{a \sqrt {-\frac {1}{b}}}\right )-a +\left (c_{1} -x \right ) b}{a}}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.204 (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*}