Internal problem ID [3811]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 37
Problem number: 1118.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\sqrt {\left (y^{\prime }\right )^{2}+1}+a y^{\prime }-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.235 (sec). Leaf size: 195
dsolve(sqrt(1+diff(y(x),x)^2)+a*diff(y(x),x) = x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x \sqrt {a^{2}+x^{2}-1}}{2 \left (a -1\right ) \left (a +1\right )}+\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right ) a^{2}}{2 \left (a -1\right ) \left (a +1\right )}-\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{2 \left (a -1\right ) \left (a +1\right )}+\frac {a \,x^{2}}{2 \left (a -1\right ) \left (a +1\right )}+c_{1} \\ y \relax (x ) = -\frac {x \sqrt {a^{2}+x^{2}-1}}{2 \left (a -1\right ) \left (a +1\right )}-\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right ) a^{2}}{2 \left (a -1\right ) \left (a +1\right )}+\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{2 \left (a -1\right ) \left (a +1\right )}+\frac {a \,x^{2}}{2 \left (a -1\right ) \left (a +1\right )}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.085 (sec). Leaf size: 113
DSolve[Sqrt[1+(y'[x])^2]+ a y'[x]==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\frac {x \left (\sqrt {a^2+x^2-1}+a x\right )}{a^2-1}-\log \left (\sqrt {a^2+x^2-1}-x\right )\right )+c_1 \\ y(x)\to \frac {1}{2} \left (\frac {x \left (a x-\sqrt {a^2+x^2-1}\right )}{a^2-1}+\log \left (\sqrt {a^2+x^2-1}-x\right )\right )+c_1 \\ \end{align*}