ODE
\[ x^{3/2} y''(x)=f\left (\frac {y(x)}{\sqrt {x}}\right ) \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 4.60693 (sec), leaf count = 572
\[\left \{\text {Solve}\left [\int _1^x\left (\frac {\sqrt {\frac {y(x)^2}{K[4]}+4 c_1+8 \int _1^{\frac {y(x)}{\sqrt {K[4]}}}f(K[2])dK[2]} y(x)}{\sqrt {K[4]} \left (y(x)^2+4 K[4] \left (c_1+2 \int _1^{\frac {y(x)}{\sqrt {K[4]}}}f(K[2])dK[2]\right )\right )}+\int _1^{y(x)}\frac {4 f\left (\frac {K[3]}{\sqrt {K[4]}}\right ) K[3]-4 \sqrt {K[4]} \left (c_1+2 \int _1^{\frac {K[3]}{\sqrt {K[4]}}}f(K[2])dK[2]\right )}{K[4] \sqrt {\frac {K[3]^2}{K[4]}+4 c_1+8 \int _1^{\frac {K[3]}{\sqrt {K[4]}}}f(K[2])dK[2]} \left (K[3]^2+4 c_1 K[4]+8 K[4] \int _1^{\frac {K[3]}{\sqrt {K[4]}}}f(K[2])dK[2]\right )}dK[3]-\frac {1}{K[4]}\right )dK[4]+c_2=\int _1^{y(x)}\frac {2}{\sqrt {x} \sqrt {\frac {K[3]^2}{x}+4 c_1+8 \int _1^{\frac {K[3]}{\sqrt {x}}}f(K[2])dK[2]}}dK[3],y(x)\right ],\text {Solve}\left [\int _1^x\left (-\frac {\sqrt {\frac {y(x)^2}{K[6]}+4 c_1+8 \int _1^{\frac {y(x)}{\sqrt {K[6]}}}f(K[2])dK[2]} y(x)}{\sqrt {K[6]} \left (y(x)^2+4 K[6] \left (c_1+2 \int _1^{\frac {y(x)}{\sqrt {K[6]}}}f(K[2])dK[2]\right )\right )}+\int _1^{y(x)}\frac {4 \sqrt {K[6]} \left (c_1+2 \int _1^{\frac {K[5]}{\sqrt {K[6]}}}f(K[2])dK[2]\right )-4 f\left (\frac {K[5]}{\sqrt {K[6]}}\right ) K[5]}{K[6] \sqrt {\frac {K[5]^2}{K[6]}+4 c_1+8 \int _1^{\frac {K[5]}{\sqrt {K[6]}}}f(K[2])dK[2]} \left (K[5]^2+4 c_1 K[6]+8 K[6] \int _1^{\frac {K[5]}{\sqrt {K[6]}}}f(K[2])dK[2]\right )}dK[5]-\frac {1}{K[6]}\right )dK[6]+c_2=\int _1^{y(x)}-\frac {2}{\sqrt {x} \sqrt {\frac {K[5]^2}{x}+4 c_1+8 \int _1^{\frac {K[5]}{\sqrt {x}}}f(K[2])dK[2]}}dK[5],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.763 (sec), leaf count = 88
\[\left [y \left (x \right ) = \RootOf \left (4 f \left (\frac {\textit {\_Z}}{\sqrt {x}}\right ) \sqrt {x}+\textit {\_Z} \right ), y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {\textit {\_C1} +8 \left (\int f \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 \textit {\_C2} \right ) \sqrt {x}, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {\textit {\_C1} +8 \left (\int f \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 \textit {\_C2} \right ) \sqrt {x}\right ]\] Mathematica raw input
DSolve[x^(3/2)*y''[x] == f[y[x]/Sqrt[x]],y[x],x]
Mathematica raw output
{Solve[C[2] + Inactive[Integrate][-K[4]^(-1) + (y[x]*Sqrt[4*C[1] + y[x]^2/K[4] + 8*Inactive[Integrate][f[K[2]], {K[2], 1, y[x]/Sqrt[K[4]]}]])/(Sqrt[K[4]]*(y[x]^2 + 4*K[4]*(C[1] + 2*Inactive[Integrate][f[K[2]], {K[2], 1, y[x]/Sqrt[K[4]]}]))) + Inactive[Integrate][(4*f[K[3]/Sqrt[K[4]]]*K[3] - 4*Sqrt[K[4]]*(C[1] + 2*Inactive[Integrate][f[K[2]], {K[2], 1, K[3]/Sqrt[K[4]]}]))/(K[4]*Sqrt[4*C[1] + K[3]^2/K[4] + 8*Inactive[Integrate][f[K[2]], {K[2], 1, K[3]/Sqrt[K[4]]}]]*(K[3]^2 + 4*C[1]*K[4] + 8*K[4]*Inactive[Integrate][f[K[2]], {K[2], 1, K[3]/Sqrt[K[4]]}])), {K[3], 1, y[x]}], {K[4], 1, x}] == Inactive[Integrate][2/(Sqrt[x]*Sqrt[4*C[1] + K[3]^2/x + 8*Inactive[Integrate][f[K[2]], {K[2], 1, K[3]/Sqrt[x]}]]), {K[3], 1, y[x]}], y[x]], Solve[C[2] + Inactive[Integrate][-K[6]^(-1) - (y[x]*Sqrt[4*C[1] + y[x]^2/K[6] + 8*Inactive[Integrate][f[K[2]], {K[2], 1, y[x]/Sqrt[K[6]]}]])/(Sqrt[K[6]]*(y[x]^2 + 4*K[6]*(C[1] + 2*Inactive[Integrate][f[K[2]], {K[2], 1, y[x]/Sqrt[K[6]]}]))) + Inactive[Integrate][(-4*f[K[5]/Sqrt[K[6]]]*K[5] + 4*Sqrt[K[6]]*(C[1] + 2*Inactive[Integrate][f[K[2]], {K[2], 1, K[5]/Sqrt[K[6]]}]))/(K[6]*Sqrt[4*C[1] + K[5]^2/K[6] + 8*Inactive[Integrate][f[K[2]], {K[2], 1, K[5]/Sqrt[K[6]]}]]*(K[5]^2 + 4*C[1]*K[6] + 8*K[6]*Inactive[Integrate][f[K[2]], {K[2], 1, K[5]/Sqrt[K[6]]}])), {K[5], 1, y[x]}], {K[6], 1, x}] == Inactive[Integrate][-2/(Sqrt[x]*Sqrt[4*C[1] + K[5]^2/x + 8*Inactive[Integrate][f[K[2]], {K[2], 1, K[5]/Sqrt[x]}]]), {K[5], 1, y[x]}], y[x]]}
Maple raw input
dsolve(x^(3/2)*diff(diff(y(x),x),x) = f(y(x)/x^(1/2)), y(x))
Maple raw output
[y(x) = RootOf(4*f(_Z/x^(1/2))*x^(1/2)+_Z), y(x) = RootOf(-ln(x)+2*Intat(1/(_C1+8*Int(f(_g),_g)+_g^2)^(1/2),_g = _Z)+2*_C2)*x^(1/2), y(x) = RootOf(-ln(x)-2*Intat(1/(_C1+8*Int(f(_g),_g)+_g^2)^(1/2),_g = _Z)+2*_C2)*x^(1/2)]