60.2.329 problem 906

Internal problem ID [10916]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 906
Date solved : Tuesday, January 28, 2025 at 05:30:55 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {x \left (1+x^{2}+y^{2}\right )}{-y^{3}-x^{2} y-y+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}} \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 37

dsolve(diff(y(x),x) = x*(x^2+y(x)^2+1)/(-y(x)^3-x^2*y(x)-y(x)+y(x)^6+3*x^2*y(x)^4+3*x^4*y(x)^2+x^6),y(x), singsol=all)
 

\[ -\frac {1}{2 x^{2}+2 y^{2}}-\frac {1}{4 \left (x^{2}+y^{2}\right )^{2}}-y+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.161 (sec). Leaf size: 127

DSolve[D[y[x],x] == (x*(1 + x^2 + y[x]^2))/(x^6 - y[x] - x^2*y[x] + 3*x^4*y[x]^2 - y[x]^3 + 3*x^2*y[x]^4 + y[x]^6),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{\left (x^2+K[2]^2\right )^2}-\frac {K[2]}{\left (x^2+K[2]^2\right )^3}-\int _1^x\left (\frac {4 K[1] K[2]}{\left (K[1]^2+K[2]^2\right )^3}+\frac {6 K[1] K[2]}{\left (K[1]^2+K[2]^2\right )^4}\right )dK[1]+1\right )dK[2]+\int _1^x\left (-\frac {K[1]}{\left (K[1]^2+y(x)^2\right )^2}-\frac {K[1]}{\left (K[1]^2+y(x)^2\right )^3}\right )dK[1]=c_1,y(x)\right ] \]