29.37.5 problem 1118

Internal problem ID [5663]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1118
Date solved : Monday, January 27, 2025 at 01:02:19 PM
CAS classification : [_quadrature]

\begin{align*} \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }&=x \end{align*}

✓ Solution by Maple

Time used: 0.088 (sec). Leaf size: 113

dsolve(sqrt(1+diff(y(x),x)^2)+a*diff(y(x),x) = x,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {a \,x^{2}+x \sqrt {a^{2}+x^{2}-1}+\left (a -1\right ) \left (a +1\right ) \left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )+2 c_{1} \right )}{2 a^{2}-2} \\ y \left (x \right ) &= \frac {a \,x^{2}-x \sqrt {a^{2}+x^{2}-1}-\left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )-2 c_{1} \right ) \left (a -1\right ) \left (a +1\right )}{2 a^{2}-2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.225 (sec). Leaf size: 154

DSolve[Sqrt[1+(D[y[x],x])^2]+ a*D[y[x],x]==x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\left (a^2-1\right )^{3/2} \left (-\sqrt {\frac {a^2+x^2-1}{a^2-1}}\right ) \text {arcsinh}\left (\frac {x}{\sqrt {a^2-1}}\right )-x \left (-a x \sqrt {a^2+x^2-1}+a^2+x^2-1\right )}{2 \left (a^2-1\right ) \sqrt {a^2+x^2-1}}+c_1 \\ 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 \\ \end{align*}