29.28.20 problem 818

Internal problem ID [5401]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 28
Problem number : 818
Date solved : Monday, January 27, 2025 at 11:18:59 AM
CAS classification : [_dAlembert]

\begin{align*} {y^{\prime }}^{2}-a y y^{\prime }-a x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.051 (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 (y \left (x \right )^{2} a +4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {a y \left (x \right )}{2}-\frac {\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \left (x \right )^{2}-2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\, a y \left (x \right )+4 a x +4}\, a} &= 0 \\ x -\frac {\left (a y \left (x \right )+\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\right ) \left (c_{1} a +\operatorname {arcsinh}\left (\frac {a y \left (x \right )}{2}+\frac {\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y \left (x \right )^{2}+2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\, a y \left (x \right )+4 a x +4}\, a} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.627 (sec). Leaf size: 61

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

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