60.2.390 problem 967

Internal problem ID [10977]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 967
Date solved : Monday, January 27, 2025 at 10:36:52 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

\begin{align*} y^{\prime }&=-\frac {x \left (-513-432 x -456 x^{6}-576 x^{5}-756 x^{3}+432 x^{3} y^{2}-216 x^{6} y^{3}+64 x^{9}-864 x^{4}-144 x^{7}+864 y^{2} x^{5}-648 y^{3} x^{4}-594 x^{2} y-378 y-1134 x^{2}-648 x^{2} y^{3}-216 y^{3}+432 y^{2} x^{7}-540 y^{2}-972 x^{4} y^{2}-96 x^{8}-288 y x^{6}-288 y x^{8}+288 y x^{7}-216 y x^{4}+720 x^{3} y-1296 x^{2} y^{2}-216 y^{2} x^{6}+1008 x^{5} y\right )}{216 \left (x^{2}+1\right )^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 91

dsolve(diff(y(x),x) = -1/216*x/(x^2+1)^4*(-513-432*x-288*y(x)*x^8+288*y(x)*x^7-288*y(x)*x^6+864*y(x)^2*x^5-648*y(x)^3*x^4-216*y(x)*x^4-456*x^6-576*x^5+432*y(x)^2*x^7-216*y(x)^2*x^6+1008*x^5*y(x)-216*x^6*y(x)^3-972*x^4*y(x)^2+432*x^3*y(x)^2+720*x^3*y(x)-648*x^2*y(x)^3-1296*x^2*y(x)^2-594*x^2*y(x)-864*x^4-378*y(x)-216*y(x)^3-756*x^3-540*y(x)^2-1134*x^2-144*x^7+64*x^9-96*x^8),y(x), singsol=all)
 

\[ y = \frac {58 \operatorname {RootOf}\left (-162 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+\ln \left (x^{2}+1\right )+6 c_{1} \right ) x^{2}+12 x^{3}-6 x^{2}+58 \operatorname {RootOf}\left (-162 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+\ln \left (x^{2}+1\right )+6 c_{1} \right )-15}{18 x^{2}+18} \]

✓ Solution by Mathematica

Time used: 1.480 (sec). Leaf size: 129

DSolve[D[y[x],x] == -1/216*(x*(-513 - 432*x - 1134*x^2 - 756*x^3 - 864*x^4 - 576*x^5 - 456*x^6 - 144*x^7 - 96*x^8 + 64*x^9 - 378*y[x] - 594*x^2*y[x] + 720*x^3*y[x] - 216*x^4*y[x] + 1008*x^5*y[x] - 288*x^6*y[x] + 288*x^7*y[x] - 288*x^8*y[x] - 540*y[x]^2 - 1296*x^2*y[x]^2 + 432*x^3*y[x]^2 - 972*x^4*y[x]^2 + 864*x^5*y[x]^2 - 216*x^6*y[x]^2 + 432*x^7*y[x]^2 - 216*y[x]^3 - 648*x^2*y[x]^3 - 648*x^4*y[x]^3 - 216*x^6*y[x]^3))/(1 + x^2)^4,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {-4 x^4+2 x^3+5 x}{2 \left (x^2+1\right )^2}+\frac {3 x y(x)}{x^2+1}}{\sqrt [3]{29} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^3}}}}\frac {1}{K[1]^3-\frac {3 K[1]}{29^{2/3}}+1}dK[1]=\frac {29^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^3}\right )^{2/3} \left (x^2+1\right )^2 \log \left (x^2+1\right )}{18 x^2}+c_1,y(x)\right ] \]