Internal problem ID [3548]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 28
Problem number: 818.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {{y^{\prime }}^{2}-a y y^{\prime }-a x=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 183
dsolve(diff(y(x),x)^2-a*y(x)*diff(y(x),x)-a*x = 0,y(x), singsol=all)
\begin{align*} x +\frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \left (a c_{1} +\operatorname {arcsinh}\left (\frac {a y \left (x \right )}{2}-\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \left (x \right )^{2}-2 a y \left (x \right ) \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4 a x +4}\, a} = 0 \\ x -\frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \left (a c_{1} +\operatorname {arcsinh}\left (\frac {a y \left (x \right )}{2}+\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \left (x \right )^{2}+2 a y \left (x \right ) \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4 a x +4}\, a} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.897 (sec). Leaf size: 75
DSolve[(y'[x])^2-a y[x] y'[x]-a x==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {K[1] \log \left (\sqrt {K[1]^2+1}-K[1]\right )}{a \sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=\frac {K[1]}{a}-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]