4.10.43 \((5 y(x)+7 x) y'(x)+8 y(x)+10 x=0\)

ODE
\[ (5 y(x)+7 x) y'(x)+8 y(x)+10 x=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0284206 (sec), leaf count = 276

\[\left \{\left \{y(x)\to \text {Root}\left [\text {$\#$1}^5+8 \text {$\#$1}^4 x+25 \text {$\#$1}^3 x^2+38 \text {$\#$1}^2 x^3+28 \text {$\#$1} x^4-e^{c_1}+8 x^5\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {$\#$1}^5+8 \text {$\#$1}^4 x+25 \text {$\#$1}^3 x^2+38 \text {$\#$1}^2 x^3+28 \text {$\#$1} x^4-e^{c_1}+8 x^5\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {$\#$1}^5+8 \text {$\#$1}^4 x+25 \text {$\#$1}^3 x^2+38 \text {$\#$1}^2 x^3+28 \text {$\#$1} x^4-e^{c_1}+8 x^5\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {$\#$1}^5+8 \text {$\#$1}^4 x+25 \text {$\#$1}^3 x^2+38 \text {$\#$1}^2 x^3+28 \text {$\#$1} x^4-e^{c_1}+8 x^5\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {$\#$1}^5+8 \text {$\#$1}^4 x+25 \text {$\#$1}^3 x^2+38 \text {$\#$1}^2 x^3+28 \text {$\#$1} x^4-e^{c_1}+8 x^5\& ,5\right ]\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 35

\[ \left \{ -{\frac {3}{5}\ln \left ({\frac {2\,x+y \relax (x ) }{x}} \right ) }-{\frac {2}{5}\ln \left ({\frac {x+y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[10*x + 8*y[x] + (7*x + 5*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-E^C[1] + 8*x^5 + 28*x^4*#1 + 38*x^3*#1^2 + 25*x^2*#1^3 + 8*x*#1^
4 + #1^5 & , 1]}, {y[x] -> Root[-E^C[1] + 8*x^5 + 28*x^4*#1 + 38*x^3*#1^2 + 25*x
^2*#1^3 + 8*x*#1^4 + #1^5 & , 2]}, {y[x] -> Root[-E^C[1] + 8*x^5 + 28*x^4*#1 + 3
8*x^3*#1^2 + 25*x^2*#1^3 + 8*x*#1^4 + #1^5 & , 3]}, {y[x] -> Root[-E^C[1] + 8*x^
5 + 28*x^4*#1 + 38*x^3*#1^2 + 25*x^2*#1^3 + 8*x*#1^4 + #1^5 & , 4]}, {y[x] -> Ro
ot[-E^C[1] + 8*x^5 + 28*x^4*#1 + 38*x^3*#1^2 + 25*x^2*#1^3 + 8*x*#1^4 + #1^5 & ,
 5]}}

Maple raw input

dsolve((7*x+5*y(x))*diff(y(x),x)+10*x+8*y(x) = 0, y(x),'implicit')

Maple raw output

-3/5*ln((2*x+y(x))/x)-2/5*ln((x+y(x))/x)-ln(x)-_C1 = 0