4.10.11 \(\left (x^2-y(x)\right ) y'(x)=4 x y(x)\)

ODE
\[ \left (x^2-y(x)\right ) y'(x)=4 x y(x) \] ODE Classification

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.0963027 (sec), leaf count = 232

\[\left \{\left \{y(x)\to x^2 \left (1+\frac {2-2 i}{\frac {i \sqrt {2}}{\sqrt {x^2 \sinh \left (\frac {2 c_1}{9}\right )+x^2 \cosh \left (\frac {2 c_1}{9}\right )-i}}-(1-i)}\right )\right \},\left \{y(x)\to x^2 \left (1+\frac {2-2 i}{(-1+i)-\frac {i \sqrt {2}}{\sqrt {x^2 \sinh \left (\frac {2 c_1}{9}\right )+x^2 \cosh \left (\frac {2 c_1}{9}\right )-i}}}\right )\right \},\left \{y(x)\to x^2 \left (1+\frac {2-2 i}{(-1+i)-\frac {\sqrt {2}}{\sqrt {x^2 \sinh \left (\frac {2 c_1}{9}\right )+x^2 \cosh \left (\frac {2 c_1}{9}\right )+i}}}\right )\right \},\left \{y(x)\to x^2 \left (1+\frac {2-2 i}{\frac {\sqrt {2}}{\sqrt {x^2 \sinh \left (\frac {2 c_1}{9}\right )+x^2 \cosh \left (\frac {2 c_1}{9}\right )+i}}-(1-i)}\right )\right \}\right \}\]

Maple
cpu = 0.02 (sec), leaf count = 29

\[ \left \{ \ln \relax (x ) -{\it \_C1}+\ln \left ({\frac {{x}^{2}+y \relax (x ) }{{x}^{2}}} \right ) -{\frac {1}{2}\ln \left ({\frac {y \relax (x ) }{{x}^{2}}} \right ) }=0 \right \} \] Mathematica raw input

DSolve[(x^2 - y[x])*y'[x] == 4*x*y[x],y[x],x]

Mathematica raw output

{{y[x] -> x^2*(1 + (2 - 2*I)/((-1 + I) + (I*Sqrt[2])/Sqrt[-I + x^2*Cosh[(2*C[1])
/9] + x^2*Sinh[(2*C[1])/9]]))}, {y[x] -> x^2*(1 + (2 - 2*I)/((-1 + I) - (I*Sqrt[
2])/Sqrt[-I + x^2*Cosh[(2*C[1])/9] + x^2*Sinh[(2*C[1])/9]]))}, {y[x] -> x^2*(1 +
 (2 - 2*I)/((-1 + I) - Sqrt[2]/Sqrt[I + x^2*Cosh[(2*C[1])/9] + x^2*Sinh[(2*C[1])
/9]]))}, {y[x] -> x^2*(1 + (2 - 2*I)/((-1 + I) + Sqrt[2]/Sqrt[I + x^2*Cosh[(2*C[
1])/9] + x^2*Sinh[(2*C[1])/9]]))}}

Maple raw input

dsolve((x^2-y(x))*diff(y(x),x) = 4*x*y(x), y(x),'implicit')

Maple raw output

ln(x)-_C1+ln((x^2+y(x))/x^2)-1/2*ln(y(x)/x^2) = 0