ODE
\[ -x^2-2 x y(x) y'(x)+3 y(x)^2 y'(x)^2+4 y(x)^2=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(x\)
Mathematica ✓
cpu = 0.188695 (sec), leaf count = 203
\[\left \{\left \{y(x)\to -\frac {\sqrt {-4 i x \sinh \left (3 c_1\right )-4 i x \cosh \left (3 c_1\right )+\sinh \left (6 c_1\right )+\cosh \left (6 c_1\right )-3 x^2}}{\sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {-4 i x \sinh \left (3 c_1\right )-4 i x \cosh \left (3 c_1\right )+\sinh \left (6 c_1\right )+\cosh \left (6 c_1\right )-3 x^2}}{\sqrt {3}}\right \},\left \{y(x)\to -\frac {\sqrt {4 i x \sinh \left (3 c_1\right )+4 i x \cosh \left (3 c_1\right )+\sinh \left (6 c_1\right )+\cosh \left (6 c_1\right )-3 x^2}}{\sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {4 i x \sinh \left (3 c_1\right )+4 i x \cosh \left (3 c_1\right )+\sinh \left (6 c_1\right )+\cosh \left (6 c_1\right )-3 x^2}}{\sqrt {3}}\right \}\right \}\]
Maple ✓
cpu = 0.705 (sec), leaf count = 197
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}-{\frac {{x}^{2}}{3}}=0,\ln \left ( x \right ) -{\frac {\sqrt {3}}{6}\sqrt {{\frac { \left ( \sqrt {3}x-3\,y \left ( x \right ) \right ) \left ( \sqrt {3}x+3\,y \left ( x \right ) \right ) }{{x}^{2}}}}}+{\frac {1}{2}\sqrt {{\frac {{x}^{2}-3\, \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}}-{\it Artanh} \left ( {\frac {1}{2}\sqrt {{\frac {{x}^{2}-3\, \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}} \right ) +{\frac {1}{2}\ln \left ( {\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\it \_C1}=0,\ln \left ( x \right ) +{\frac {\sqrt {3}}{6}\sqrt {{\frac { \left ( \sqrt {3}x-3\,y \left ( x \right ) \right ) \left ( \sqrt {3}x+3\,y \left ( x \right ) \right ) }{{x}^{2}}}}}-{\frac {1}{2}\sqrt {{\frac {{x}^{2}-3\, \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}}+{\it Artanh} \left ( {\frac {1}{2}\sqrt {{\frac {{x}^{2}-3\, \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}} \right ) +{\frac {1}{2}\ln \left ( {\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[-x^2 + 4*y[x]^2 - 2*x*y[x]*y'[x] + 3*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[-3*x^2 - (4*I)*x*Cosh[3*C[1]] + Cosh[6*C[1]] - (4*I)*x*Sinh[3*C[1]] + Sinh[6*C[1]]]/Sqrt[3])}, {y[x] -> Sqrt[-3*x^2 - (4*I)*x*Cosh[3*C[1]] + Cosh[6*C[1]] - (4*I)*x*Sinh[3*C[1]] + Sinh[6*C[1]]]/Sqrt[3]}, {y[x] -> -(Sqrt[-3*x^2 + (4*I)*x*Cosh[3*C[1]] + Cosh[6*C[1]] + (4*I)*x*Sinh[3*C[1]] + Sinh[6*C[1]]]/Sqrt[3])}, {y[x] -> Sqrt[-3*x^2 + (4*I)*x*Cosh[3*C[1]] + Cosh[6*C[1]] + (4*I)*x*Sinh[3*C[1]] + Sinh[6*C[1]]]/Sqrt[3]}}
Maple raw input
dsolve(3*y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-x^2+4*y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x)^2-1/3*x^2 = 0, ln(x)-1/6*3^(1/2)*((3^(1/2)*x-3*y(x))*(3^(1/2)*x+3*y(x))/x^2)^(1/2)+1/2*((x^2-3*y(x)^2)/x^2)^(1/2)-arctanh(1/2*((x^2-3*y(x)^2)/x^2)^(1/2))+1/2*ln((x^2+y(x)^2)/x^2)-_C1 = 0, ln(x)+1/6*3^(1/2)*((3^(1/2)*x-3*y(x))*(3^(1/2)*x+3*y(x))/x^2)^(1/2)-1/2*((x^2-3*y(x)^2)/x^2)^(1/2)+arctanh(1/2*((x^2-3*y(x)^2)/x^2)^(1/2))+1/2*ln((x^2+y(x)^2)/x^2)-_C1 = 0