\[ \sqrt {y(x)} y''(x)-a=0 \] ✓ Mathematica : cpu = 0.0736745 (sec), leaf count = 1677
DSolve[-a + Sqrt[y[x]]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {3 c_1{}^2}{16 a^2}+\frac {\sqrt [3]{-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2+\sqrt {4 \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right ){}^3+\left (-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2\right ){}^2}}}{768 \sqrt [3]{2}}-\frac {-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1}{384\ 2^{2/3} \sqrt [3]{-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2+\sqrt {4 \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right ){}^3+\left (-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2\right ){}^2}}}\right \},\left \{y(x)\to \frac {3 c_1{}^2}{16 a^2}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2+\sqrt {4 \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right ){}^3+\left (-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2\right ){}^2}}}{1536 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right )}{768\ 2^{2/3} \sqrt [3]{-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2+\sqrt {4 \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right ){}^3+\left (-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2\right ){}^2}}}\right \},\left \{y(x)\to \frac {3 c_1{}^2}{16 a^2}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2+\sqrt {4 \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right ){}^3+\left (-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2\right ){}^2}}}{1536 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right )}{768\ 2^{2/3} \sqrt [3]{-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2+\sqrt {4 \left (-\frac {2304 c_1{}^4}{a^4}-663552 x^2 c_1-663552 c_2{}^2 c_1-1327104 x c_2 c_1\right ){}^3+\left (-\frac {221184 c_1{}^6}{a^6}+\frac {159252480 x^2 c_1{}^3}{a^2}+\frac {159252480 c_2{}^2 c_1{}^3}{a^2}+\frac {318504960 x c_2 c_1{}^3}{a^2}+2293235712 a^2 x^4+2293235712 a^2 c_2{}^4+9172942848 a^2 x c_2{}^3+13759414272 a^2 x^2 c_2{}^2+9172942848 a^2 x^3 c_2\right ){}^2}}}\right \}\right \}\] ✓ Maple : cpu = 0.122 (sec), leaf count = 91
dsolve(y(x)^(1/2)*diff(diff(y(x),x),x)-a=0,y(x))
\[\frac {-3 c_{1} \sqrt {4 a \sqrt {y \left (x \right )}-c_{1}}-\left (4 a \sqrt {y \left (x \right )}-c_{1}\right )^{\frac {3}{2}}}{12 a^{2}}-x -c_{2} = 0\]