ODE
\[ \sqrt {y(x)} y''(x)=a \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.101042 (sec), leaf count = 1881
\[\left \{\left \{y(x)\to \frac {288 c_1 c_2^2 a^4+288 x^2 c_1 a^4+576 x c_1 c_2 a^4+\left (\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}\right ){}^{2/3} a^4+3 c_1^2 \sqrt [3]{\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}} a^2+c_1^4}{16 a^4 \sqrt [3]{\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}}}\right \},\left \{y(x)\to \frac {-288 i \sqrt {3} c_1 c_2^2 a^4-288 c_1 c_2^2 a^4-288 i \sqrt {3} x^2 c_1 a^4-288 x^2 c_1 a^4-576 i \sqrt {3} x c_1 c_2 a^4-576 x c_1 c_2 a^4+i \sqrt {3} \left (\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}\right ){}^{2/3} a^4-\left (\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}\right ){}^{2/3} a^4+6 c_1^2 \sqrt [3]{\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}} a^2-i \sqrt {3} c_1^4-c_1^4}{32 a^4 \sqrt [3]{\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}}}\right \},\left \{y(x)\to \frac {288 i \sqrt {3} c_1 c_2^2 a^4-288 c_1 c_2^2 a^4+288 i \sqrt {3} x^2 c_1 a^4-288 x^2 c_1 a^4+576 i \sqrt {3} x c_1 c_2 a^4-576 x c_1 c_2 a^4-i \sqrt {3} \left (\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}\right ){}^{2/3} a^4-\left (\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}\right ){}^{2/3} a^4+6 c_1^2 \sqrt [3]{\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}} a^2+i \sqrt {3} c_1^4-c_1^4}{32 a^4 \sqrt [3]{\frac {10368 x^4 a^8+10368 c_2^4 a^8+41472 x c_2^3 a^8+62208 x^2 c_2^2 a^8+41472 x^3 c_2 a^8+48 \sqrt {\frac {\left (x+c_2\right ){}^2 \left (36 a^4 \left (x+c_2\right ){}^2-c_1^3\right ){}^3}{a^8}} a^6+720 x^2 c_1^3 a^4+720 c_1^3 c_2^2 a^4+1440 x c_1^3 c_2 a^4-c_1^6}{a^6}}}\right \}\right \}\]
Maple ✓
cpu = 0.295 (sec), leaf count = 91
\[ \left \{ {\frac {1}{12\,{a}^{2}} \left ( -3\,{\it \_C1}\,\sqrt {4\,a\sqrt {y \left ( x \right ) }-{\it \_C1}}- \left ( 4\,a\sqrt {y \left ( x \right ) }-{\it \_C1} \right ) ^{{\frac {3}{2}}} \right ) }-x-{\it \_C2}=0,{\frac {1}{12\,{a}^{2}} \left ( 3\,{\it \_C1}\,\sqrt {4\,a\sqrt {y \left ( x \right ) }-{\it \_C1}}+ \left ( 4\,a\sqrt {y \left ( x \right ) }-{\it \_C1} \right ) ^{{\frac {3}{2}}} \right ) }-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[Sqrt[y[x]]*y''[x] == a,y[x],x]
Mathematica raw output
{{y[x] -> (288*a^4*x^2*C[1] + C[1]^4 + 576*a^4*x*C[1]*C[2] + 288*a^4*C[1]*C[2]^2 + 3*a^2*C[1]^2*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(1/3) + a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(2/3))/(16*a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(1/3))}, {y[x] -> (-288*a^4*x^2*C[1] - (288*I)*Sqrt[3]*a^4*x^2*C[1] - C[1]^4 - I*Sqrt[3]*C[1]^4 - 576*a^4*x*C[1]*C[2] - (576*I)*Sqrt[3]*a^4*x*C[1]*C[2] - 288*a^4*C[1]*C[2]^2 - (288*I)*Sqrt[3]*a^4*C[1]*C[2]^2 + 6*a^2*C[1]^2*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(1/3) - a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(2/3) + I*Sqrt[3]*a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(2/3))/(32*a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(1/3))}, {y[x] -> (-288*a^4*x^2*C[1] + (288*I)*Sqrt[3]*a^4*x^2*C[1] - C[1]^4 + I*Sqrt[3]*C[1]^4 - 576*a^4*x*C[1]*C[2] + (576*I)*Sqrt[3]*a^4*x*C[1]*C[2] - 288*a^4*C[1]*C[2]^2 + (288*I)*Sqrt[3]*a^4*C[1]*C[2]^2 + 6*a^2*C[1]^2*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(1/3) - a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(2/3) - I*Sqrt[3]*a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(2/3))/(32*a^4*((10368*a^8*x^4 + 720*a^4*x^2*C[1]^3 - C[1]^6 + 41472*a^8*x^3*C[2] + 1440*a^4*x*C[1]^3*C[2] + 62208*a^8*x^2*C[2]^2 + 720*a^4*C[1]^3*C[2]^2 + 41472*a^8*x*C[2]^3 + 10368*a^8*C[2]^4 + 48*a^6*Sqrt[((x + C[2])^2*(-C[1]^3 + 36*a^4*(x + C[2])^2)^3)/a^8])/a^6)^(1/3))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)*y(x)^(1/2) = a, y(x),'implicit')
Maple raw output
1/12*(-3*_C1*(4*a*y(x)^(1/2)-_C1)^(1/2)-(4*a*y(x)^(1/2)-_C1)^(3/2))/a^2-x-_C2 = 0, 1/12*(3*_C1*(4*a*y(x)^(1/2)-_C1)^(1/2)+(4*a*y(x)^(1/2)-_C1)^(3/2))/a^2-x-_C2 = 0