4.38.43 $2f(x{)}^2{y}^{″}(x)=f(x)y^{\prime }(x)(3f^{\prime }(x)-2f(x)y(x))+f(x)y(x{)}^2{f}^{\prime }(x)+y(x)(f(x)f^{″}(x)-2f^{\prime }(x{)}^2-2f(x{)}^3)+2f(x{)}^{2}y(x{)}^3$

ODE
$$2f(x{)}^2{y}^{″}(x)=f(x)y^{\prime }(x)(3f^{\prime }(x)-2f(x)y(x))+f(x)y(x{)}^2{f}^{\prime }(x)+y(x)(f(x)f^{″}(x)-2f^{\prime }(x{)}^2-2f(x{)}^3)+2f(x{)}^{2}y(x{)}^3$$ 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

$$\{\int \sqrt{f\left(x\right)}\mathrm{d}x-{\int }^{y\left(x\right)\frac{1}{\sqrt{f\left(x\right)}}}\frac{1}{2{\mathit{\_}\mathit{f}}^8-8{\mathit{\_}\mathit{f}}^6+12{\mathit{\_}\mathit{f}}^4+(-160{\mathit{\_}\mathit{C}\mathit{1}}^3-8){\mathit{\_}\mathit{f}}^2+160{\mathit{\_}\mathit{C}\mathit{1}}^3+2}(-\frac{{(\mathit{\_}\mathit{f}-1)}^3(i\sqrt{3}+1)({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1){(\mathit{\_}\mathit{f}+1)}^3{4}^{\frac{2}{3}}}{4}\frac{1}{\sqrt[3]{{(\mathit{\_}\mathit{f}-1)}^3(\sqrt{5}\mathit{\_}\mathit{C}\mathit{1}\sqrt{-\frac{\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1}}+\frac{1}{4}){({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)}^2{(\mathit{\_}\mathit{f}+1)}^3}}+(i\sqrt{3}-1)\sqrt[3]{4}\sqrt[3]{{(\mathit{\_}\mathit{f}-1)}^3(\sqrt{5}\mathit{\_}\mathit{C}\mathit{1}\sqrt{-\frac{\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1}}+\frac{1}{4}){({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)}^2{(\mathit{\_}\mathit{f}+1)}^3})d\mathit{\_}\mathit{f}-\mathit{\_}\mathit{C}\mathit{2}=0,\int \sqrt{f\left(x\right)}\mathrm{d}x-{\int }^{y\left(x\right)\frac{1}{\sqrt{f\left(x\right)}}}\frac{1}{2{\mathit{\_}\mathit{f}}^8-8{\mathit{\_}\mathit{f}}^6+12{\mathit{\_}\mathit{f}}^4+(-160{\mathit{\_}\mathit{C}\mathit{1}}^3-8){\mathit{\_}\mathit{f}}^2+160{\mathit{\_}\mathit{C}\mathit{1}}^3+2}(\frac{(i\sqrt{3}-1){(\mathit{\_}\mathit{f}-1)}^3({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1){(\mathit{\_}\mathit{f}+1)}^3{4}^{\frac{2}{3}}}{4}\frac{1}{\sqrt[3]{{(\mathit{\_}\mathit{f}-1)}^3(\sqrt{5}\mathit{\_}\mathit{C}\mathit{1}\sqrt{-\frac{\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1}}+\frac{1}{4}){({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)}^2{(\mathit{\_}\mathit{f}+1)}^3}}+(-i\sqrt{3}-1)\sqrt[3]{4}\sqrt[3]{{(\mathit{\_}\mathit{f}-1)}^3(\sqrt{5}\mathit{\_}\mathit{C}\mathit{1}\sqrt{-\frac{\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1}}+\frac{1}{4}){({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)}^2{(\mathit{\_}\mathit{f}+1)}^3})d\mathit{\_}\mathit{f}-\mathit{\_}\mathit{C}\mathit{2}=0,\int \sqrt{f\left(x\right)}\mathrm{d}x-{\int }^{y\left(x\right)\frac{1}{\sqrt{f\left(x\right)}}}\frac{1}{({\mathit{\_}\mathit{f}}^2-1)({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)}(\frac{{(\mathit{\_}\mathit{f}-1)}^3{(\mathit{\_}\mathit{f}+1)}^3({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)4^{\frac{2}{3}}}{4}\frac{1}{\sqrt[3]{{(\mathit{\_}\mathit{f}-1)}^3(\sqrt{5}\mathit{\_}\mathit{C}\mathit{1}\sqrt{-\frac{\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1}}+\frac{1}{4}){({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)}^2{(\mathit{\_}\mathit{f}+1)}^3}}+\sqrt[3]{4}\sqrt[3]{{(\mathit{\_}\mathit{f}-1)}^3(\sqrt{5}\mathit{\_}\mathit{C}\mathit{1}\sqrt{-\frac{\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1}}+\frac{1}{4}){({\mathit{\_}\mathit{f}}^6-3{\mathit{\_}\mathit{f}}^4-80{\mathit{\_}\mathit{C}\mathit{1}}^3+3{\mathit{\_}\mathit{f}}^2-1)}^2{(\mathit{\_}\mathit{f}+1)}^3})d\mathit{\_}\mathit{f}-\mathit{\_}\mathit{C}\mathit{2}=0,y\left(x\right)=0,y\left(x\right)=\sqrt{f\left(x\right)},y\left(x\right)=-\sqrt{f\left(x\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