Internal problem ID [9389]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1811 (book 6.220).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {\sqrt {y}\, y^{\prime \prime }-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 91
dsolve(y(x)^(1/2)*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
\begin{align*} -\frac {\frac {\left (4 a \sqrt {y \relax (x )}-c_{1}\right )^{\frac {3}{2}}}{3}+c_{1} \sqrt {4 a \sqrt {y \relax (x )}-c_{1}}}{4 a^{2}}-x -c_{2} = 0 \\ \frac {\frac {\left (4 a \sqrt {y \relax (x )}-c_{1}\right )^{\frac {3}{2}}}{3}+c_{1} \sqrt {4 a \sqrt {y \relax (x )}-c_{1}}}{4 a^{2}}-x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.055 (sec). Leaf size: 910
DSolve[-a + Sqrt[y[x]]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a^4 \left (\left (\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}\right ){}^{2/3}+288 c_1 (x+c_2){}^2\right )+3 a^2 c_1{}^2 \sqrt [3]{\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}}+c_1{}^4}{16 a^4 \sqrt [3]{\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}}} \\ y(x)\to \frac {i a^4 \left (\left (\sqrt {3}+i\right ) \left (\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}\right ){}^{2/3}-288 \left (\sqrt {3}-i\right ) c_1 (x+c_2){}^2\right )+6 a^2 c_1{}^2 \sqrt [3]{\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}}+\left (-1-i \sqrt {3}\right ) c_1{}^4}{32 a^4 \sqrt [3]{\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}}} \\ y(x)\to \frac {-i a^4 \left (\left (\sqrt {3}-i\right ) \left (\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}\right ){}^{2/3}-288 \left (\sqrt {3}+i\right ) c_1 (x+c_2){}^2\right )+6 a^2 c_1{}^2 \sqrt [3]{\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}}+i \left (\sqrt {3}+i\right ) c_1{}^4}{32 a^4 \sqrt [3]{\frac {10368 a^8 (x+c_2){}^4+720 a^4 c_1{}^3 (x+c_2){}^2+48 a^6 \sqrt {\frac {(x+c_2){}^2 \left (36 a^4 (x+c_2){}^2-c_1{}^3\right ){}^3}{a^8}}-c_1{}^6}{a^6}}} \\ \end{align*}