4.15.2 $(-7xy(x{)}^3-y(x)+5x)y^{\prime }(x)-y(x{)}^4+5y(x)=0$

ODE
$$(-7xy(x{)}^3-y(x)+5x)y^{\prime }(x)-y(x{)}^4+5y(x)=0$$ ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0212264 (sec), leaf count = 302

$$\{\{y(x)\to \text{Root}[10{\mathrm{\#}\text{1}}^{7}x+2{\mathrm{\#}\text{1}}^5-100{\mathrm{\#}\text{1}}^{4}x-25{\mathrm{\#}\text{1}}^2+250\mathrm{\#}\text{1}x-10c_1\mathrm{\&},1]\},\{y(x)\to \text{Root}[10{\mathrm{\#}\text{1}}^{7}x+2{\mathrm{\#}\text{1}}^5-100{\mathrm{\#}\text{1}}^{4}x-25{\mathrm{\#}\text{1}}^2+250\mathrm{\#}\text{1}x-10c_1\mathrm{\&},2]\},\{y(x)\to \text{Root}[10{\mathrm{\#}\text{1}}^{7}x+2{\mathrm{\#}\text{1}}^5-100{\mathrm{\#}\text{1}}^{4}x-25{\mathrm{\#}\text{1}}^2+250\mathrm{\#}\text{1}x-10c_1\mathrm{\&},3]\},\{y(x)\to \text{Root}[10{\mathrm{\#}\text{1}}^{7}x+2{\mathrm{\#}\text{1}}^5-100{\mathrm{\#}\text{1}}^{4}x-25{\mathrm{\#}\text{1}}^2+250\mathrm{\#}\text{1}x-10c_1\mathrm{\&},4]\},\{y(x)\to \text{Root}[10{\mathrm{\#}\text{1}}^{7}x+2{\mathrm{\#}\text{1}}^5-100{\mathrm{\#}\text{1}}^{4}x-25{\mathrm{\#}\text{1}}^2+250\mathrm{\#}\text{1}x-10c_1\mathrm{\&},5]\},\{y(x)\to \text{Root}[10{\mathrm{\#}\text{1}}^{7}x+2{\mathrm{\#}\text{1}}^5-100{\mathrm{\#}\text{1}}^{4}x-25{\mathrm{\#}\text{1}}^2+250\mathrm{\#}\text{1}x-10c_1\mathrm{\&},6]\},\{y(x)\to \text{Root}[10{\mathrm{\#}\text{1}}^{7}x+2{\mathrm{\#}\text{1}}^5-100{\mathrm{\#}\text{1}}^{4}x-25{\mathrm{\#}\text{1}}^2+250\mathrm{\#}\text{1}x-10c_1\mathrm{\&},7]\}\}$$

Maple ✓
cpu = 0.016 (sec), leaf count = 35

$$\{x+\frac{2{\left(y\left(x\right)\right)}^5-25{\left(y\left(x\right)\right)}^2-10\mathit{\_}\mathit{C}\mathit{1}}{10y\left(x\right){({\left(y\left(x\right)\right)}^3-5)}^2}=0\}$$ Mathematica raw input

DSolve[5*y[x] - y[x]^4 + (5*x - y[x] - 7*x*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 + 2*#1^5 + 10*x*#1^7 &
 , 1]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 + 2*#1^5 + 10*x
*#1^7 & , 2]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 + 2*#1^5
 + 10*x*#1^7 & , 3]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 +
 2*#1^5 + 10*x*#1^7 & , 4]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x
*#1^4 + 2*#1^5 + 10*x*#1^7 & , 5]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 
- 100*x*#1^4 + 2*#1^5 + 10*x*#1^7 & , 6]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 2
5*#1^2 - 100*x*#1^4 + 2*#1^5 + 10*x*#1^7 & , 7]}}

Maple raw input

dsolve((5*x-y(x)-7*x*y(x)^3)*diff(y(x),x)+5*y(x)-y(x)^4 = 0, y(x),'implicit')

Maple raw output

x+1/10*(2*y(x)^5-25*y(x)^2-10*_C1)/y(x)/(y(x)^3-5)^2 = 0