ODE
\[ -4 a^2 x y'(x)+a^2 y(x)+y(x) y'(x)^2=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 8.614 (sec), leaf count = 749
\[\left \{\text {Solve}\left [\frac {8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )+\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )-2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )+\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )-8 \tan ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}+\log (x)=c_1,y(x)\right ],\text {Solve}\left [\frac {\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (-4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )+2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )+\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )+8 \tan ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )\right )-8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}+\log (x)=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.194 (sec), leaf count = 181
\[\left [-\frac {\textit {\_C1} x}{y \left (x \right ) a \left (\frac {\left (2 a x +\sqrt {4 a^{2} x^{2}-y \left (x \right )^{2}}\right ) a}{y \left (x \right )}\right )^{\frac {1}{3}} \left (\frac {a^{2} \left (2 a^{2} x^{2}+\sqrt {4 a^{2} x^{2}-y \left (x \right )^{2}}\, a x -y \left (x \right )^{2}\right )}{y \left (x \right )^{2}}\right )^{\frac {1}{3}}}+x = 0, -\frac {\textit {\_C1} x}{y \left (x \right ) a \left (-\frac {\left (-2 a x +\sqrt {4 a^{2} x^{2}-y \left (x \right )^{2}}\right ) a}{y \left (x \right )}\right )^{\frac {1}{3}} \left (-\frac {4 a^{2} \left (-2 a^{2} x^{2}+\sqrt {4 a^{2} x^{2}-y \left (x \right )^{2}}\, a x +y \left (x \right )^{2}\right )}{y \left (x \right )^{2}}\right )^{\frac {1}{3}}}+x = 0\right ]\] Mathematica raw input
DSolve[a^2*y[x] - 4*a^2*x*y'[x] + y[x]*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[Log[x] + (8*ArcSinh[Sqrt[-2*a + y[x]/x]/(2*Sqrt[a])]*(4*a^2 - y[x]^2/x^2)^(3/2) + Sqrt[a]*Sqrt[2 + y[x]/(a*x)]*((-8*ArcTan[Sqrt[2*a - y[x]/x]/Sqrt[2*a + y[x]/x]] + 4*Log[y[x]/x] + Log[3*a^2 - y[x]^2/x^2])*Sqrt[-(-2*a + y[x]/x)^2]*Sqrt[2*a + y[x]/x]*Sqrt[4*a^2 - y[x]^2/x^2] + 4*ArcTanh[Sqrt[4*a^2 - y[x]^2/x^2]/(2*a)]*Sqrt[-2*a + y[x]/x]*(-4*a^2 + y[x]^2/x^2) - 2*ArcTanh[Sqrt[4*a^2 - y[x]^2/x^2]/a]*Sqrt[-2*a + y[x]/x]*(-4*a^2 + y[x]^2/x^2)))/(6*Sqrt[a]*Sqrt[-(-2*a + y[x]/x)^2]*Sqrt[2*a + y[x]/x]*Sqrt[2 + y[x]/(a*x)]*Sqrt[4*a^2 - y[x]^2/x^2]) == C[1], y[x]], Solve[Log[x] + (-8*ArcSinh[Sqrt[-2*a + y[x]/x]/(2*Sqrt[a])]*(4*a^2 - y[x]^2/x^2)^(3/2) + Sqrt[a]*Sqrt[2 + y[x]/(a*x)]*((8*ArcTan[Sqrt[2*a - y[x]/x]/Sqrt[2*a + y[x]/x]] + 4*Log[y[x]/x] + Log[3*a^2 - y[x]^2/x^2])*Sqrt[-(-2*a + y[x]/x)^2]*Sqrt[2*a + y[x]/x]*Sqrt[4*a^2 - y[x]^2/x^2] - 4*ArcTanh[Sqrt[4*a^2 - y[x]^2/x^2]/(2*a)]*Sqrt[-2*a + y[x]/x]*(-4*a^2 + y[x]^2/x^2) + 2*ArcTanh[Sqrt[4*a^2 - y[x]^2/x^2]/a]*Sqrt[-2*a + y[x]/x]*(-4*a^2 + y[x]^2/x^2)))/(6*Sqrt[a]*Sqrt[-(-2*a + y[x]/x)^2]*Sqrt[2*a + y[x]/x]*Sqrt[2 + y[x]/(a*x)]*Sqrt[4*a^2 - y[x]^2/x^2]) == C[1], y[x]]}
Maple raw input
dsolve(y(x)*diff(y(x),x)^2-4*a^2*x*diff(y(x),x)+a^2*y(x) = 0, y(x))
Maple raw output
[-_C1*x/y(x)/a/((2*a*x+(4*a^2*x^2-y(x)^2)^(1/2))/y(x)*a)^(1/3)/(a^2*(2*a^2*x^2+(4*a^2*x^2-y(x)^2)^(1/2)*a*x-y(x)^2)/y(x)^2)^(1/3)+x = 0, -_C1*x/y(x)/a/(-(-2*a*x+(4*a^2*x^2-y(x)^2)^(1/2))/y(x)*a)^(1/3)/(-4*a^2*(-2*a^2*x^2+(4*a^2*x^2-y(x)^2)^(1/2)*a*x+y(x)^2)/y(x)^2)^(1/3)+x = 0]