60.7.220 problem 1811 (book 6.220)

Internal problem ID [11809]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1811 (book 6.220)
Date solved : Monday, January 27, 2025 at 11:36:58 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} \sqrt {y}\, y^{\prime \prime }-a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.138 (sec). Leaf size: 81

dsolve(y(x)^(1/2)*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
 

\begin{align*} \frac {\left (-2 a \sqrt {y}-c_{1} \right ) \sqrt {4 a \sqrt {y}-c_{1}}-6 a^{2} \left (x +c_{2} \right )}{6 a^{2}} &= 0 \\ \frac {\left (2 a \sqrt {y}+c_{1} \right ) \sqrt {4 a \sqrt {y}-c_{1}}-6 a^{2} \left (x +c_{2} \right )}{6 a^{2}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.116 (sec). Leaf size: 1881

DSolve[-a + Sqrt[y[x]]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {288 a^4 c_1 x^2+576 a^4 c_1 c_2 x+288 a^4 c_1 c_2{}^2+a^4 \left (\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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}+3 a^2 c_1{}^2 \sqrt [3]{\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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 {-288 i \sqrt {3} a^4 c_1 x^2-288 a^4 c_1 x^2-576 i \sqrt {3} a^4 c_1 c_2 x-576 a^4 c_1 c_2 x-288 i \sqrt {3} a^4 c_1 c_2{}^2-288 a^4 c_1 c_2{}^2+i \sqrt {3} a^4 \left (\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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}-a^4 \left (\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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}+6 a^2 c_1{}^2 \sqrt [3]{\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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 \sqrt {3} c_1{}^4-c_1{}^4}{32 a^4 \sqrt [3]{\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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 {288 i \sqrt {3} a^4 c_1 x^2-288 a^4 c_1 x^2+576 i \sqrt {3} a^4 c_1 c_2 x-576 a^4 c_1 c_2 x+288 i \sqrt {3} a^4 c_1 c_2{}^2-288 a^4 c_1 c_2{}^2-i \sqrt {3} a^4 \left (\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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}-a^4 \left (\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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}+6 a^2 c_1{}^2 \sqrt [3]{\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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 \sqrt {3} c_1{}^4-c_1{}^4}{32 a^4 \sqrt [3]{\frac {10368 a^8 x^4+41472 a^8 c_2 x^3+62208 a^8 c_2{}^2 x^2+41472 a^8 c_2{}^3 x+10368 a^8 c_2{}^4+720 a^4 c_1{}^3 x^2+1440 a^4 c_1{}^3 c_2 x+720 a^4 c_1{}^3 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*}