4.18.49 \(x^2 y'(x)^2-y(x)^4+y(x)^2=0\)

ODE
\[ x^2 y'(x)^2-y(x)^4+y(x)^2=0 \] ODE Classification

[_separable]

Book solution method
No Missing Variables ODE, Solve for \(y'\)

Mathematica
cpu = 0.0408584 (sec), leaf count = 103

\[\left \{\left \{y(x)\to \sqrt {\sec ^2\left (c_1-\log (x)\right )} \left (-\cot \left (c_1-\log (x)\right )\right )\right \},\left \{y(x)\to \sqrt {\sec ^2\left (c_1-\log (x)\right )} \cot \left (c_1-\log (x)\right )\right \},\left \{y(x)\to \sqrt {\sec ^2\left (c_1+\log (x)\right )} \left (-\cot \left (c_1+\log (x)\right )\right )\right \},\left \{y(x)\to \sqrt {\sec ^2\left (c_1+\log (x)\right )} \cot \left (c_1+\log (x)\right )\right \}\right \}\]

Maple
cpu = 0.077 (sec), leaf count = 45

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

DSolve[y[x]^2 - y[x]^4 + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(Cot[C[1] - Log[x]]*Sqrt[Sec[C[1] - Log[x]]^2])}, {y[x] -> Cot[C[1] -
 Log[x]]*Sqrt[Sec[C[1] - Log[x]]^2]}, {y[x] -> -(Cot[C[1] + Log[x]]*Sqrt[Sec[C[1
] + Log[x]]^2])}, {y[x] -> Cot[C[1] + Log[x]]*Sqrt[Sec[C[1] + Log[x]]^2]}}

Maple raw input

dsolve(x^2*diff(y(x),x)^2+y(x)^2-y(x)^4 = 0, y(x),'implicit')

Maple raw output

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