60.2.367 problem 944

Internal problem ID [10954]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 944
Date solved : Monday, January 27, 2025 at 10:30:47 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]

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

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 46

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

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

✓ Solution by Mathematica

Time used: 0.550 (sec). Leaf size: 129

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

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