Internal problem ID [4056]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 28
Problem number: 817.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {{y^{\prime }}^{2}+a y y^{\prime }=x a} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 394
dsolve(diff(y(x),x)^2+a*y(x)*diff(y(x),x)-a*x = 0,y(x), singsol=all)
\begin{align*} \frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) c_{1}}{\sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}-4}\, \sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4}}+x +\frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \ln \left (-\frac {a y \left (x \right )}{2}+\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}+\frac {\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}}{2}\right )}{a \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}} = 0 \frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) c_{1}}{\sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}-4}\, \sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4}}+x -\frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \ln \left (-\frac {a y \left (x \right )}{2}-\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}+\frac {\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}}{2}\right )}{a \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}} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 0.633 (sec). Leaf size: 83
DSolve[(y'[x])^2+a y[x] y'[x]-a x==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {2 K[1] \arctan \left (\frac {\sqrt {1-K[1]^2}}{K[1]+1}\right )}{a \sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-\frac {K[1]}{a}\right \},\{y(x),K[1]\}\right ] \]