60.2.368 problem 945

Internal problem ID [10955]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 945
Date solved : Monday, January 27, 2025 at 10:30:49 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {-32 y x -8 x^{3}-16 a \,x^{2}-32 x +64 y^{3}+48 x^{2} y^{2}+96 a x y^{2}+12 y x^{4}+48 y a \,x^{3}+48 a^{2} x^{2} y+x^{6}+6 x^{5} a +12 a^{2} x^{4}+8 a^{3} x^{3}}{64 y+16 x^{2}+32 a x +64} \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 42

dsolve(diff(y(x),x) = (-32*x*y(x)-8*x^3-16*a*x^2-32*x+64*y(x)^3+48*x^2*y(x)^2+96*a*x*y(x)^2+12*y(x)*x^4+48*y(x)*a*x^3+48*a^2*x^2*y(x)+x^6+6*x^5*a+12*a^2*x^4+8*a^3*x^3)/(64*y(x)+16*x^2+32*a*x+64),y(x), singsol=all)
 

\[ y = -\frac {x^{2}}{4}-\frac {a x}{2}+\operatorname {RootOf}\left (-x +2 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a} +1}{2 \textit {\_a}^{3}+\textit {\_a} a +a}d \textit {\_a} \right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.463 (sec). Leaf size: 127

DSolve[D[y[x],x] == (-32*x - 16*a*x^2 - 8*x^3 + 8*a^3*x^3 + 12*a^2*x^4 + 6*a*x^5 + x^6 - 32*x*y[x] + 48*a^2*x^2*y[x] + 48*a*x^3*y[x] + 12*x^4*y[x] + 96*a*x*y[x]^2 + 48*x^2*y[x]^2 + 64*y[x]^3)/(64 + 32*a*x + 16*x^2 + 64*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {768}{x^2+2 a x+4 y(x)+4}-32 (a+6)}{32 \sqrt [3]{2} \sqrt [3]{-a \left (a^2+18 a+54\right )}}}\frac {1}{K[1]^3+\frac {3 \sqrt [3]{-1} \sqrt [3]{a} (a+12) K[1]}{2^{2/3} \left (a^2+18 a+54\right )^{2/3}}+1}dK[1]=\frac {\left (-a \left (a^2+18 a+54\right )\right )^{2/3} x}{18 \sqrt [3]{2}}+c_1,y(x)\right ] \]