4.36.43 $y^{″}(x)=y(x)(f^{\prime }(x)-2f(x{)}^2)+(3f(x)-y(x))y^{\prime }(x)+f(x)y(x{)}^2+y(x{)}^3$

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

(ODEtools/info) missing specification of intermediate function

Book solution method
TO DO

Mathematica ✗
cpu = 0.455828 (sec), leaf count = 0 , could not solve

DSolve[Derivative[2][y][x] == f[x]*y[x]^2 + y[x]^3 + y[x]*(-2*f[x]^2 + Derivative[1][f][x]) + (3*f[x] - y[x])*Derivative[1][y][x], y[x], x]

Maple ✓
cpu = 0.783 (sec), leaf count = 442

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

DSolve[y''[x] == f[x]*y[x]^2 + y[x]^3 + y[x]*(-2*f[x]^2 + f'[x]) + (3*f[x] - y[x])*y'[x],y[x],x]

Mathematica raw output

DSolve[Derivative[2][y][x] == f[x]*y[x]^2 + y[x]^3 + y[x]*(-2*f[x]^2 + Derivativ
e[1][f][x]) + (3*f[x] - y[x])*Derivative[1][y][x], y[x], x]

Maple raw input

dsolve(diff(diff(y(x),x),x) = (3*f(x)-y(x))*diff(y(x),x)+(diff(f(x),x)-2*f(x)^2)*y(x)+f(x)*y(x)^2+y(x)^3, y(x),'implicit')

Maple raw output

y(x) = 0, y(x) = 1/2*(-1-(5+4*exp(Int(f(x),x))*exp(-Int(f(x),x)))^(1/2))/exp(-In
t(f(x),x))/(Int(exp(Int(f(x),x)),x)+_C3), y(x) = 1/2*(-1+(5+4*exp(Int(f(x),x))*e
xp(-Int(f(x),x)))^(1/2))/exp(-Int(f(x),x))/(Int(exp(Int(f(x),x)),x)+_C3), Int(ex
p(Int(f(x),x)),x)-Intat(1/(-_f^6+_C1)*((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2))
)^(1/3)-_f^2/((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3),_f = y(x)*exp(-In
t(f(x),x)))-_C2 = 0, Int(exp(Int(f(x),x)),x)-Intat((-_f^2*(I*3^(1/2)-1)*(_f^6-_C
1)/((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3)+((_f^6-_C1)^2*(-1+(_C1/(-_f
^6+_C1))^(1/2)))^(1/3)*(I*3^(1/2)+1))/(2*_f^6-2*_C1),_f = y(x)*exp(-Int(f(x),x))
)-_C2 = 0, Int(exp(Int(f(x),x)),x)-Intat((_f^2*(I*3^(1/2)+1)*(_f^6-_C1)/((_f^6-_
C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3)+(-I*3^(1/2)+1)*((_f^6-_C1)^2*(-1+(_C1/
(-_f^6+_C1))^(1/2)))^(1/3))/(2*_f^6-2*_C1),_f = y(x)*exp(-Int(f(x),x)))-_C2 = 0