75.7.23 problem 198

Internal problem ID [16816]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 198
Date solved : Tuesday, January 28, 2025 at 09:33:43 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 55

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

\begin{align*} y &= \frac {2 c_{1} +1-\sqrt {2 c_{1} x^{2}+2 c_{1} +1}}{2 c_{1}} \\ y &= \frac {2 c_{1} +1+\sqrt {2 c_{1} x^{2}+2 c_{1} +1}}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.282 (sec). Leaf size: 295

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

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