4.3.44 \(3 y'(x)=\sqrt {x^2-3 y(x)}+x\)

ODE
\[ 3 y'(x)=\sqrt {x^2-3 y(x)}+x \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.144187 (sec), leaf count = 547

\[\left \{\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,6\right ]\right \}\right \}\]

Maple
cpu = 0.091 (sec), leaf count = 131

\[ \left \{ -{\frac {1}{y \relax (x ) } \left (\left (x-\sqrt {{x}^{2}-3\,y \relax (x ) } \right ) \sqrt {2\,\sqrt {{x}^{2}-3\,y \relax (x ) }+x}+\sqrt {2\,\sqrt {{x}^{2}-3\,y \relax (x ) }-x}\sqrt {-{x}^{2}+4\,y \relax (x ) }y \relax (x ) {\it \_C1}\, \left (x+\sqrt {{x}^{2}-3\,y \relax (x ) } \right ) \right ) {\frac {1}{\sqrt {2\,\sqrt {{x}^{2}-3\,y \relax (x ) }-x}}}{\frac {1}{\sqrt {-{x}^{2}+4\,y \relax (x ) }}} \left (x+\sqrt {{x}^{2}-3\,y \relax (x ) } \right ) ^{-1}}=0 \right \} \] Mathematica raw input

DSolve[3*y'[x] == x + Sqrt[x^2 - 3*y[x]],y[x],x]

Mathematica raw output

{{y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E
^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#
1^6 & , 1]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^
4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#
1^5 + 3888*#1^6 & , 2]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E
^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 
- 1944*x^2*#1^5 + 3888*#1^6 & , 3]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])
*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 2
43*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 4]}, {y[x] -> Root[3*E^(12*C[1]) - 1
6*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[
1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 5]}, {y[x] -> Root[3*E^(
12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 +
 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 6]}}

Maple raw input

dsolve(3*diff(y(x),x) = x+(x^2-3*y(x))^(1/2), y(x),'implicit')

Maple raw output

-1/(2*(x^2-3*y(x))^(1/2)-x)^(1/2)/(-x^2+4*y(x))^(1/2)*((x-(x^2-3*y(x))^(1/2))*(2
*(x^2-3*y(x))^(1/2)+x)^(1/2)+(2*(x^2-3*y(x))^(1/2)-x)^(1/2)*(-x^2+4*y(x))^(1/2)*
y(x)*_C1*(x+(x^2-3*y(x))^(1/2)))/(x+(x^2-3*y(x))^(1/2))/y(x) = 0