4.13.10 \(\left (3 x^2-y(x)^2\right ) y'(x)=2 x y(x)\)

ODE
\[ \left (3 x^2-y(x)^2\right ) y'(x)=2 x y(x) \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.178843 (sec), leaf count = 458

\[\left \{\left \{y(x)\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right )\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3}\right \},\left \{y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3}\right \}\right \}\]

Maple
cpu = 0.025 (sec), leaf count = 40

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

DSolve[(3*x^2 - y[x]^2)*y'[x] == 2*x*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (-E^C[1] + (2^(1/3)*E^(2*C[1]))/(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sqr
t[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3) + (-2*E^(3*C[1]) + 27*E^C[1]*x^
2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3)/2^(1/3))/3}, {y[x] -> 
-E^C[1]/3 - ((I/3)*(-I + Sqrt[3])*E^(2*C[1]))/(2^(2/3)*(-2*E^(3*C[1]) + 27*E^C[1
]*x^2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3)) + ((I/6)*(I + Sqr
t[3])*(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1])
*x^4])^(1/3))/2^(1/3)}, {y[x] -> -E^C[1]/3 + ((I/3)*(I + Sqrt[3])*E^(2*C[1]))/(2
^(2/3)*(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sqrt[-12*E^(4*C[1])*x^2 + 81*E^(2*C[1]
)*x^4])^(1/3)) - ((I/6)*(-I + Sqrt[3])*(-2*E^(3*C[1]) + 27*E^C[1]*x^2 + 3*Sqrt[-
12*E^(4*C[1])*x^2 + 81*E^(2*C[1])*x^4])^(1/3))/2^(1/3)}}

Maple raw input

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

Maple raw output

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