60.7.64 problem 1655 (book 6.64)

Internal problem ID [11653]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1655 (book 6.64)
Date solved : Monday, January 27, 2025 at 11:29:21 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Solution by Maple

Time used: 0.258 (sec). Leaf size: 55

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

✓ Solution by Mathematica

Time used: 60.686 (sec). Leaf size: 308

DSolve[-2*a*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 c_2-\frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}} \\ y(x)\to \frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}}+c_2 \\ \end{align*}