75.12.12 problem 286

Internal problem ID [16878]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 286
Date solved : Tuesday, January 28, 2025 at 09:38:50 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 4.556 (sec). Leaf size: 40

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

\[ y = \left (\operatorname {RootOf}\left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}-\textit {\_Z}^{4}-2 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x \]

✓ Solution by Mathematica

Time used: 0.170 (sec). Leaf size: 50

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]^2-K[1]+1}{(K[1]-2) (K[1]-1) K[1]}dK[1]=-\log (x)+c_1,y(x)\right ] \]