60.2.43 problem 619

Internal problem ID [10630]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 619
Date solved : Tuesday, January 28, 2025 at 05:00:23 PM
CAS classification : [`x=_G(y,y')`]

\begin{align*} y^{\prime }&=\frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 81

dsolve(diff(y(x),x) = 6*y(x)/(8*y(x)^4+9*y(x)^3+12*y(x)^2+6*y(x)-F(-1/3*y(x)^4-1/2*y(x)^3-y(x)^2-y(x)+x)),y(x), singsol=all)
 

\[ \int _{\textit {\_b}}^{y}\frac {-8 \textit {\_a}^{4}-9 \textit {\_a}^{3}-12 \textit {\_a}^{2}+F \left (-\frac {1}{3} \textit {\_a}^{4}-\frac {1}{2} \textit {\_a}^{3}-\textit {\_a}^{2}-\textit {\_a} +x \right )-6 \textit {\_a}}{F \left (-\frac {1}{3} \textit {\_a}^{4}-\frac {1}{2} \textit {\_a}^{3}-\textit {\_a}^{2}-\textit {\_a} +x \right ) \textit {\_a}}d \textit {\_a} -c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.438 (sec). Leaf size: 330

DSolve[D[y[x],x] == (6*y[x])/(-F[x - y[x] - y[x]^2 - y[x]^3/2 - y[x]^4/3] + 6*y[x] + 12*y[x]^2 + 9*y[x]^3 + 8*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {8 K[2]^3}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {9 K[2]^2}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {12 K[2]}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right ) \int _1^x-\frac {6 \left (-\frac {4}{3} K[2]^3-\frac {3 K[2]^2}{2}-2 K[2]-1\right ) F''\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+K[1]\right )}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+K[1]\right )^2}dK[1]+6}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}+\frac {1}{K[2]}\right )dK[2]+\int _1^x\frac {6}{F\left (-\frac {1}{3} y(x)^4-\frac {y(x)^3}{2}-y(x)^2-y(x)+K[1]\right )}dK[1]=c_1,y(x)\right ] \]