(−16y(x) + 7x + 140)y′(x) + y(x) + 8x + 25 = 0

4.10.50 $(-16 y(x)+7 x+140) y'(x)+y(x)+8 x+25=0$

ODE
\[ (-16 y(x)+7 x+140) y'(x)+y(x)+8 x+25=0 \] ODE Classiﬁcation

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

Book solution method
Equation linear in the variables, $y'(x)=f\left ( \frac {X_1}{X_2} \right ) $

Mathematica ✓
cpu = 0.10437 (sec), leaf count = 1673

\[\left \{\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,1\right ]}\right )\right \},\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,2\right ]}\right )\right \},\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,3\right ]}\right )\right \},\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,4\right ]}\right )\right \},\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,5\right ]}\right )\right \},\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,6\right ]}\right )\right \},\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,7\right ]}\right )\right \},\left \{y(x)\to \frac {1}{16} \left (7 (x+20)-\frac {1}{\text {Root}\left [\left (553584375 x^8+17714700000 x^7+248005800000 x^6+1984046400000 x^5+9920232000000 x^4+31744742400000 x^3+63489484800000 x^2+72559411200000 x+553584375 e^{30 c_1}+36279705600000\right ) \text {$\#$1}^8+\left (-16402500 x^6-393660000 x^5-3936600000 x^4-20995200000 x^3-62985600000 x^2-100776960000 x-67184640000\right ) \text {$\#$1}^6+\left (486000 x^5+9720000 x^4+77760000 x^3+311040000 x^2+622080000 x+497664000\right ) \text {$\#$1}^5+\left (166050 x^4+2656800 x^3+15940800 x^2+42508800 x+42508800\right ) \text {$\#$1}^4+\left (-9504 x^3-114048 x^2-456192 x-608256\right ) \text {$\#$1}^3+\left (-468 x^2-3744 x-7488\right ) \text {$\#$1}^2+(48 x+192) \text {$\#$1}-1\& ,8\right ]}\right )\right \}\right \}\]

Maple ✓
cpu = 0.03 (sec), leaf count = 47

\[ \left \{ -{\frac {5}{8}\ln \left ( {\frac {10-2\,y \left ( x \right ) -x}{4+x}} \right ) }-{\frac {3}{8}\ln \left ( {\frac {11-y \left ( x \right ) +x}{4+x}} \right ) }-\ln \left ( 4+x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[25 + 8*x + y[x] + (140 + 7*x - 16*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*x - 468*x^2)*#
1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (42508800 + 42508800*x
+ 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 622080000*x + 31
1040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-67184640000 - 10
0776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4 - 393660000*x^
5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) + 7255941120000
0*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4 + 198404640000
0*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 & , 1]^(-1))/16
}, {y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*x - 468*x^2)
*#1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (42508800 + 42508800
*x + 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 622080000*x +
311040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-67184640000 -
100776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4 - 393660000*
x^5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) + 72559411200
000*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4 + 1984046400
000*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 & , 2]^(-1))/
16}, {y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*x - 468*x^
2)*#1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (42508800 + 425088
00*x + 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 622080000*x
+ 311040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-67184640000
- 100776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4 - 39366000
0*x^5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) + 725594112
00000*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4 + 19840464
00000*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 & , 3]^(-1)
)/16}, {y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*x - 468*
x^2)*#1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (42508800 + 4250
8800*x + 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 622080000*
x + 311040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-6718464000
0 - 100776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4 - 393660
000*x^5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) + 7255941
1200000*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4 + 198404
6400000*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 & , 4]^(-
1))/16}, {y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*x - 46
8*x^2)*#1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (42508800 + 42
508800*x + 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 62208000
0*x + 311040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-67184640
000 - 100776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4 - 3936
60000*x^5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) + 72559
411200000*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4 + 1984
046400000*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 & , 5]^
(-1))/16}, {y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*x -
468*x^2)*#1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (42508800 +
42508800*x + 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 622080
000*x + 311040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-671846
40000 - 100776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4 - 39
3660000*x^5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) + 725
59411200000*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4 + 19
84046400000*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 & , 6
]^(-1))/16}, {y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*x
- 468*x^2)*#1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (42508800
+ 42508800*x + 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 6220
80000*x + 311040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-6718
4640000 - 100776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4 -
393660000*x^5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) + 7
2559411200000*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4 +
1984046400000*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 & ,
7]^(-1))/16}, {y[x] -> (7*(20 + x) - Root[-1 + (192 + 48*x)*#1 + (-7488 - 3744*
x - 468*x^2)*#1^2 + (-608256 - 456192*x - 114048*x^2 - 9504*x^3)*#1^3 + (4250880
0 + 42508800*x + 15940800*x^2 + 2656800*x^3 + 166050*x^4)*#1^4 + (497664000 + 62
2080000*x + 311040000*x^2 + 77760000*x^3 + 9720000*x^4 + 486000*x^5)*#1^5 + (-67
184640000 - 100776960000*x - 62985600000*x^2 - 20995200000*x^3 - 3936600000*x^4
- 393660000*x^5 - 16402500*x^6)*#1^6 + (36279705600000 + 553584375*E^(30*C[1]) +
72559411200000*x + 63489484800000*x^2 + 31744742400000*x^3 + 9920232000000*x^4
+ 1984046400000*x^5 + 248005800000*x^6 + 17714700000*x^7 + 553584375*x^8)*#1^8 &
, 8]^(-1))/16}}

Maple raw input

dsolve((140+7*x-16*y(x))*diff(y(x),x)+25+8*x+y(x) = 0, y(x),'implicit')

Maple raw output

-5/8*ln((10-2*y(x)-x)/(4+x))-3/8*ln((11-y(x)+x)/(4+x))-ln(4+x)-_C1 = 0

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4.10.50 \((-16 y(x)+7 x+140) y'(x)+y(x)+8 x+25=0\)