60.7.65 problem 1656 (book 6.65)

Internal problem ID [11654]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1656 (book 6.65)
Date solved : Monday, January 27, 2025 at 11:29:24 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }-a y \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}}&=0 \end{align*}

Solution by Maple

Time used: 0.284 (sec). Leaf size: 102

dsolve(diff(diff(y(x),x),x)-a*y(x)*(diff(y(x),x)^2+1)^(3/2)=0,y(x), singsol=all)
 
\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ a \left (\int _{}^{y}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_{1} \right )^{2} a^{2}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_{1} \right )^{2} a^{2}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

Solution by Mathematica

Time used: 1.052 (sec). Leaf size: 1104

DSolve[-(a*y[x]*(1 + D[y[x],x]^2)^(3/2)) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 (-1) c_1}{-1-c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 (-1) c_1}{1-c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right ),\frac {1-c_1}{-c_1-1}\right )+(-1-c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right )|\frac {1-c_1}{-c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 (-1) c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a (-c_1)-4+4 (-c_1){}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 (-1) c_1}{-1-c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 (-1) c_1}{1-c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right ),\frac {1-c_1}{-c_1-1}\right )+(-1-c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 (-1) c_1+2}} \text {$\#$1}\right )|\frac {1-c_1}{-c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 (-1) c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a (-c_1)-4+4 (-c_1){}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ \end{align*}