ODE
\[ a^2 x+y(x)^3 \left (-y'(x)\right )+x y(x)^2 y'(x)^2=0 \] ODE Classification
[[_homogeneous, `class G`], _rational]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.345739 (sec), leaf count = 132
\[\left \{\left \{y(x)\to -\sqrt {2 a^2 e^{c_1}+\frac {1}{2} e^{-c_1} x^2}\right \},\left \{y(x)\to \sqrt {2 a^2 e^{c_1}+\frac {1}{2} e^{-c_1} x^2}\right \},\left \{y(x)\to -\frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}}\right \}\right \}\]
Maple ✓
cpu = 1.575 (sec), leaf count = 161
\[ \left \{ {1 \left ( \ln \left ( {\frac {1}{x} \left ( 2\, \left ( y \left ( x \right ) \right ) ^{4}+2\,\sqrt { \left ( y \left ( x \right ) \right ) ^{4}}\sqrt { \left ( y \left ( x \right ) \right ) ^{4}-4\,{a}^{2}{x}^{2}} \right ) } \right ) \left ( y \left ( x \right ) \right ) ^{2}+ \left ( {\it \_C1}-\ln \left ( x \right ) \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{4}}-2\, \left ( y \left ( x \right ) \right ) ^{2}\ln \left ( y \left ( x \right ) \right ) \right ) {\frac {1}{\sqrt { \left ( y \left ( x \right ) \right ) ^{4}}}}}=0,-{1 \left ( \ln \left ( {\frac {1}{x} \left ( 2\, \left ( y \left ( x \right ) \right ) ^{4}+2\,\sqrt { \left ( y \left ( x \right ) \right ) ^{4}}\sqrt { \left ( y \left ( x \right ) \right ) ^{4}-4\,{a}^{2}{x}^{2}} \right ) } \right ) \left ( y \left ( x \right ) \right ) ^{2}+ \left ( -{\it \_C1}+\ln \left ( x \right ) \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{4}}-2\, \left ( y \left ( x \right ) \right ) ^{2}\ln \left ( y \left ( x \right ) \right ) \right ) {\frac {1}{\sqrt { \left ( y \left ( x \right ) \right ) ^{4}}}}}=0, \left ( y \left ( x \right ) \right ) ^{4}-4\,{a}^{2}{x}^{2}=0 \right \} \] Mathematica raw input
DSolve[a^2*x - y[x]^3*y'[x] + x*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[2*a^2*E^C[1] + x^2/(2*E^C[1])]}, {y[x] -> Sqrt[2*a^2*E^C[1] + x^2/(2*E^C[1])]}, {y[x] -> -(Sqrt[E^C[1] + (4*a^2*x^2)/E^C[1]]/Sqrt[2])}, {y[x] -> Sqrt[E^C[1] + (4*a^2*x^2)/E^C[1]]/Sqrt[2]}}
Maple raw input
dsolve(x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)+a^2*x = 0, y(x),'implicit')
Maple raw output
y(x)^4-4*a^2*x^2 = 0, -(ln((2*y(x)^4+2*(y(x)^4)^(1/2)*(y(x)^4-4*a^2*x^2)^(1/2))/x)*y(x)^2+(-_C1+ln(x))*(y(x)^4)^(1/2)-2*y(x)^2*ln(y(x)))/(y(x)^4)^(1/2) = 0, (ln((2*y(x)^4+2*(y(x)^4)^(1/2)*(y(x)^4-4*a^2*x^2)^(1/2))/x)*y(x)^2+(_C1-ln(x))*(y(x)^4)^(1/2)-2*y(x)^2*ln(y(x)))/(y(x)^4)^(1/2) = 0