ODE
\[ 2 f(x)^2 y''(x)=f(x) y'(x) \left (3 f'(x)-2 f(x) y(x)\right )+f(x) y(x)^2 f'(x)+y(x) \left (f(x) f''(x)-2 f'(x)^2-2 f(x)^3\right )+2 f(x)^2 y(x)^3 \] ODE Classification
(ODEtools/info) missing specification of intermediate function
Book solution method
TO DO
Mathematica ✗
cpu = 0.984603 (sec), leaf count = 0 , could not solve
DSolve[2*f[x]^2*Derivative[2][y][x] == 2*f[x]^2*y[x]^3 + f[x]*y[x]^2*Derivative[1][f][x] + f[x]*(-2*f[x]*y[x] + 3*Derivative[1][f][x])*Derivative[1][y][x] + y[x]*(-2*f[x]^3 - 2*Derivative[1][f][x]^2 + f[x]*Derivative[2][f][x]), y[x], x]
Maple ✓
cpu = 2.31 (sec), leaf count = 761
\[ \left \{ \int \!\sqrt {f \left ( x \right ) }\,{\rm d}x-\int ^{{y \left ( x \right ) {\frac {1}{\sqrt {f \left ( x \right ) }}}}}\!{\frac {1}{2\,{{\it \_f}}^{8}-8\,{{\it \_f}}^{6}+12\,{{\it \_f}}^{4}+ \left ( -160\,{{\it \_C1}}^{3}-8 \right ) {{\it \_f}}^{2}+160\,{{\it \_C1}}^{3}+2} \left ( -{\frac { \left ( {\it \_f}-1 \right ) ^{3} \left ( i\sqrt {3}+1 \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) \left ( {\it \_f}+1 \right ) ^{3}{4}^{{\frac {2}{3}}}}{4}{\frac {1}{\sqrt [3]{ \left ( {\it \_f}-1 \right ) ^{3} \left ( \sqrt {5}{\it \_C1}\,\sqrt {-{\frac {{\it \_C1}}{{{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1}}}+{\frac {1}{4}} \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) ^{2} \left ( {\it \_f}+1 \right ) ^{3}}}}}+ \left ( i\sqrt {3}-1 \right ) \sqrt [3]{4}\sqrt [3]{ \left ( {\it \_f}-1 \right ) ^{3} \left ( \sqrt {5}{\it \_C1}\,\sqrt {-{\frac {{\it \_C1}}{{{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1}}}+{\frac {1}{4}} \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) ^{2} \left ( {\it \_f}+1 \right ) ^{3}} \right ) }{d{\it \_f}}-{\it \_C2}=0,\int \!\sqrt {f \left ( x \right ) }\,{\rm d}x-\int ^{{y \left ( x \right ) {\frac {1}{\sqrt {f \left ( x \right ) }}}}}\!{\frac {1}{2\,{{\it \_f}}^{8}-8\,{{\it \_f}}^{6}+12\,{{\it \_f}}^{4}+ \left ( -160\,{{\it \_C1}}^{3}-8 \right ) {{\it \_f}}^{2}+160\,{{\it \_C1}}^{3}+2} \left ( {\frac { \left ( i\sqrt {3}-1 \right ) \left ( {\it \_f}-1 \right ) ^{3} \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) \left ( {\it \_f}+1 \right ) ^{3}{4}^{{\frac {2}{3}}}}{4}{\frac {1}{\sqrt [3]{ \left ( {\it \_f}-1 \right ) ^{3} \left ( \sqrt {5}{\it \_C1}\,\sqrt {-{\frac {{\it \_C1}}{{{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1}}}+{\frac {1}{4}} \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) ^{2} \left ( {\it \_f}+1 \right ) ^{3}}}}}+ \left ( -i\sqrt {3}-1 \right ) \sqrt [3]{4}\sqrt [3]{ \left ( {\it \_f}-1 \right ) ^{3} \left ( \sqrt {5}{\it \_C1}\,\sqrt {-{\frac {{\it \_C1}}{{{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1}}}+{\frac {1}{4}} \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) ^{2} \left ( {\it \_f}+1 \right ) ^{3}} \right ) }{d{\it \_f}}-{\it \_C2}=0,\int \!\sqrt {f \left ( x \right ) }\,{\rm d}x-\int ^{{y \left ( x \right ) {\frac {1}{\sqrt {f \left ( x \right ) }}}}}\!{\frac {1}{ \left ( {{\it \_f}}^{2}-1 \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) } \left ( {\frac { \left ( {\it \_f}-1 \right ) ^{3} \left ( {\it \_f}+1 \right ) ^{3} \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) {4}^{{\frac {2}{3}}}}{4}{\frac {1}{\sqrt [3]{ \left ( {\it \_f}-1 \right ) ^{3} \left ( \sqrt {5}{\it \_C1}\,\sqrt {-{\frac {{\it \_C1}}{{{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1}}}+{\frac {1}{4}} \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) ^{2} \left ( {\it \_f}+1 \right ) ^{3}}}}}+\sqrt [3]{4}\sqrt [3]{ \left ( {\it \_f}-1 \right ) ^{3} \left ( \sqrt {5}{\it \_C1}\,\sqrt {-{\frac {{\it \_C1}}{{{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1}}}+{\frac {1}{4}} \right ) \left ( {{\it \_f}}^{6}-3\,{{\it \_f}}^{4}-80\,{{\it \_C1}}^{3}+3\,{{\it \_f}}^{2}-1 \right ) ^{2} \left ( {\it \_f}+1 \right ) ^{3}} \right ) }{d{\it \_f}}-{\it \_C2}=0,y \left ( x \right ) =0,y \left ( x \right ) =\sqrt {f \left ( x \right ) },y \left ( x \right ) =-\sqrt {f \left ( x \right ) } \right \} \] Mathematica raw input
DSolve[2*f[x]^2*y''[x] == 2*f[x]^2*y[x]^3 + f[x]*y[x]^2*f'[x] + f[x]*(-2*f[x]*y[x] + 3*f'[x])*y'[x] + y[x]*(-2*f[x]^3 - 2*f'[x]^2 + f[x]*f''[x]),y[x],x]
Mathematica raw output
DSolve[2*f[x]^2*Derivative[2][y][x] == 2*f[x]^2*y[x]^3 + f[x]*y[x]^2*Derivative[1][f][x] + f[x]*(-2*f[x]*y[x] + 3*Derivative[1][f][x])*Derivative[1][y][x] + y[x]*(-2*f[x]^3 - 2*Derivative[1][f][x]^2 + f[x]*Derivative[2][f][x]), y[x], x]
Maple raw input
dsolve(2*f(x)^2*diff(diff(y(x),x),x) = f(x)*(3*diff(f(x),x)-2*f(x)*y(x))*diff(y(x),x)+(f(x)*diff(diff(f(x),x),x)-2*diff(f(x),x)^2-2*f(x)^3)*y(x)+f(x)*diff(f(x),x)*y(x)^2+2*f(x)^2*y(x)^3, y(x),'implicit')
Maple raw output
y(x) = 0, y(x) = f(x)^(1/2), y(x) = -f(x)^(1/2), Int(f(x)^(1/2),x)-Intat((1/4*(_f-1)^3*(_f+1)^3*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)*4^(2/3)/((_f-1)^3*(5^(1/2)*_C1*(-_C1/(_f^6-3*_f^4-80*_C1^3+3*_f^2-1))^(1/2)+1/4)*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)^2*(_f+1)^3)^(1/3)+4^(1/3)*((_f-1)^3*(5^(1/2)*_C1*(-_C1/(_f^6-3*_f^4-80*_C1^3+3*_f^2-1))^(1/2)+1/4)*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)^2*(_f+1)^3)^(1/3))/(_f^2-1)/(_f^6-3*_f^4-80*_C1^3+3*_f^2-1),_f = y(x)/f(x)^(1/2))-_C2 = 0, Int(f(x)^(1/2),x)-Intat((-1/4*(_f-1)^3*(I*3^(1/2)+1)*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)*(_f+1)^3*4^(2/3)/((_f-1)^3*(5^(1/2)*_C1*(-_C1/(_f^6-3*_f^4-80*_C1^3+3*_f^2-1))^(1/2)+1/4)*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)^2*(_f+1)^3)^(1/3)+(I*3^(1/2)-1)*4^(1/3)*((_f-1)^3*(5^(1/2)*_C1*(-_C1/(_f^6-3*_f^4-80*_C1^3+3*_f^2-1))^(1/2)+1/4)*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)^2*(_f+1)^3)^(1/3))/(2*_f^8-8*_f^6+12*_f^4+(-160*_C1^3-8)*_f^2+160*_C1^3+2),_f = y(x)/f(x)^(1/2))-_C2 = 0, Int(f(x)^(1/2),x)-Intat((1/4*(I*3^(1/2)-1)*(_f-1)^3*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)*(_f+1)^3*4^(2/3)/((_f-1)^3*(5^(1/2)*_C1*(-_C1/(_f^6-3*_f^4-80*_C1^3+3*_f^2-1))^(1/2)+1/4)*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)^2*(_f+1)^3)^(1/3)+(-I*3^(1/2)-1)*4^(1/3)*((_f-1)^3*(5^(1/2)*_C1*(-_C1/(_f^6-3*_f^4-80*_C1^3+3*_f^2-1))^(1/2)+1/4)*(_f^6-3*_f^4-80*_C1^3+3*_f^2-1)^2*(_f+1)^3)^(1/3))/(2*_f^8-8*_f^6+12*_f^4+(-160*_C1^3-8)*_f^2+160*_C1^3+2),_f = y(x)/f(x)^(1/2))-_C2 = 0