32.6.40 problem Exercise 12.40, page 103

Internal problem ID [5905]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.40, page 103
Date solved : Monday, January 27, 2025 at 01:25:33 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.132 (sec). Leaf size: 57

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

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

✓ Solution by Mathematica

Time used: 2.507 (sec). Leaf size: 246

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

\begin{align*} y(x)\to x^2 \left (1+\frac {2-2 i}{\frac {i \sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )-i}}-(1-i)}\right ) \\ y(x)\to x^2 \left (1+\frac {2-2 i}{(-1+i)-\frac {i \sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )-i}}}\right ) \\ y(x)\to x^2 \left (1+\frac {2-2 i}{(-1+i)-\frac {\sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )+i}}}\right ) \\ y(x)\to x^2 \left (1+\frac {2-2 i}{\frac {\sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )+i}}-(1-i)}\right ) \\ y(x)\to 0 \\ y(x)\to -x^2 \\ \end{align*}