60.2.195 problem 771

Internal problem ID [10782]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 771
Date solved : Monday, January 27, 2025 at 09:44:09 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }&=\frac {-4 a x y-a^{2} x^{3}-2 a \,x^{2} b -4 a x +8}{8 y+2 a \,x^{2}+4 b x +8} \end{align*}

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 87

dsolve(diff(y(x),x) = (-4*y(x)*a*x-a^2*x^3-2*a*x^2*b-4*a*x+8)/(8*y(x)+2*a*x^2+4*b*x+8),y(x), singsol=all)
 

\[ y = \frac {-a b \,x^{2}-2 x \,b^{2}-4 b +4 \,{\mathrm e}^{\frac {-4 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {\left (-x \,b^{2}-2 b -4\right ) a -2 c_{1} b^{2}}{4 a}}}{2}\right ) a +\left (-x \,b^{2}-2 b -4\right ) a -2 c_{1} b^{2}}{4 a}}-8}{4 b} \]

✓ Solution by Mathematica

Time used: 2.577 (sec). Leaf size: 76

DSolve[D[y[x],x] == (8 - 4*a*x - 2*a*b*x^2 - a^2*x^3 - 4*a*x*y[x])/(8 + 4*b*x + 2*a*x^2 + 8*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {a b x^2+8 W\left (-e^{-\frac {b^2 x}{4}-1+c_1}\right )+2 b^2 x+4 b+8}{4 b} \\ y(x)\to -\frac {a b x^2+2 b^2 x+4 b+8}{4 b} \\ \end{align*}