12.9.47 problem 39 part(e)

Internal problem ID [1803]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 39 part(e)
Date solved : Monday, January 27, 2025 at 05:35:19 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} x^{2} \left (y^{\prime }+y^{2}\right )+y x +x^{2}-\frac {1}{4}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 40

dsolve(x^2*(diff(y(x),x)+y(x)^2)+x*y(x)+x^2-1/4=0,y(x), singsol=all)
 

\[ y = \frac {-4 c_1 x -{\mathrm e}^{-2 i x}-2 i {\mathrm e}^{-2 i x} x -2 i c_1}{2 x \left ({\mathrm e}^{-2 i x}+2 i c_1 \right )} \]

✓ Solution by Mathematica

Time used: 0.378 (sec). Leaf size: 22

DSolve[x^2*(D[y[x],x]+y[x]^2)+x*y[x]+x^2-1/4==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {1}{2 x}-\tan (x-c_1) \]