60.7.61 problem 1652 (book 6.61)

Internal problem ID [11650]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1652 (book 6.61)
Date solved : Monday, January 27, 2025 at 11:29:11 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Solution by Maple

Time used: 0.128 (sec). Leaf size: 31

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

\[ y = \int \operatorname {RootOf}\left (x -\int _{}^{\textit {\_Z}}\frac {1}{a \sqrt {\textit {\_f}^{2}+1}+b}d \textit {\_f} +c_{1} \right )d x +c_{2} \]

✓ Solution by Mathematica

Time used: 1.884 (sec). Leaf size: 46

DSolve[D[y[x],{x,2}]==a*Sqrt[1+D[y[x],x]^2]+b,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {K[1]^2+1} a+b}dK[1]\&\right ][c_1+K[2]]dK[2]+c_2 \]