28.20 problem 818

Internal problem ID [3549]

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]

Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-a y y^{\prime }-a x=0} \end {gather*}

Solution by Maple

Time used: 0.085 (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 \relax (x )+\sqrt {a \left (a y \relax (x )^{2}+4 x \right )}\right ) \left (c_{1} a +\arcsinh \left (\frac {a y \relax (x )}{2}-\frac {\sqrt {a \left (a y \relax (x )^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \relax (x )^{2}-2 a y \relax (x ) \sqrt {a \left (a y \relax (x )^{2}+4 x \right )}+4 a x +4}\, a} = 0 \\ x -\frac {\left (a y \relax (x )+\sqrt {a \left (a y \relax (x )^{2}+4 x \right )}\right ) \left (c_{1} a +\arcsinh \left (\frac {a y \relax (x )}{2}+\frac {\sqrt {a \left (a y \relax (x )^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \relax (x )^{2}+2 a y \relax (x ) \sqrt {a \left (a y \relax (x )^{2}+4 x \right )}+4 a x +4}\, a} = 0 \\ \end{align*}

Solution by Mathematica

Time used: 0.476 (sec). Leaf size: 72

DSolve[(y'[x])^2-a y[x] y'[x]-a x==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{x=\frac {K[1] \tanh ^{-1}\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )}{a \sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=\frac {K[1]}{a}-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]