##### 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 = 0.385683 (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 = 0.077 (sec), leaf count = 278

$\left [-\frac {2 y \left (x \right )^{2} \sqrt {x^{2}-3 y \left (x \right )}\, x^{4}}{\left (\sqrt {x^{2}-3 y \left (x \right )}-x \right )^{2} \left (2 \sqrt {x^{2}-3 y \left (x \right )}+x \right )}-\frac {2 y \left (x \right )^{2} x^{5}}{\left (\sqrt {x^{2}-3 y \left (x \right )}-x \right )^{2} \left (2 \sqrt {x^{2}-3 y \left (x \right )}+x \right )}+\frac {17 y \left (x \right )^{3} x^{3}}{\left (\sqrt {x^{2}-3 y \left (x \right )}-x \right )^{2} \left (2 \sqrt {x^{2}-3 y \left (x \right )}+x \right )}+\frac {14 y \left (x \right )^{3} \sqrt {x^{2}-3 y \left (x \right )}\, x^{2}}{\left (\sqrt {x^{2}-3 y \left (x \right )}-x \right )^{2} \left (2 \sqrt {x^{2}-3 y \left (x \right )}+x \right )}-\frac {36 y \left (x \right )^{4} x}{\left (\sqrt {x^{2}-3 y \left (x \right )}-x \right )^{2} \left (2 \sqrt {x^{2}-3 y \left (x \right )}+x \right )}-\frac {24 y \left (x \right )^{4} \sqrt {x^{2}-3 y \left (x \right )}}{\left (\sqrt {x^{2}-3 y \left (x \right )}-x \right )^{2} \left (2 \sqrt {x^{2}-3 y \left (x \right )}+x \right )}-\textit {\_C1} = 0\right ]$ 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))

Maple raw output

[-2*y(x)^2/((x^2-3*y(x))^(1/2)-x)^2/(2*(x^2-3*y(x))^(1/2)+x)*(x^2-3*y(x))^(1/2)*
x^4-2*y(x)^2/((x^2-3*y(x))^(1/2)-x)^2/(2*(x^2-3*y(x))^(1/2)+x)*x^5+17*y(x)^3/((x
^2-3*y(x))^(1/2)-x)^2/(2*(x^2-3*y(x))^(1/2)+x)*x^3+14*y(x)^3/((x^2-3*y(x))^(1/2)
-x)^2/(2*(x^2-3*y(x))^(1/2)+x)*(x^2-3*y(x))^(1/2)*x^2-36*y(x)^4/((x^2-3*y(x))^(1
/2)-x)^2/(2*(x^2-3*y(x))^(1/2)+x)*x-24*y(x)^4/((x^2-3*y(x))^(1/2)-x)^2/(2*(x^2-3
*y(x))^(1/2)+x)*(x^2-3*y(x))^(1/2)-_C1 = 0]