60.2.323 problem 900

Internal problem ID [10910]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 900
Date solved : Monday, January 27, 2025 at 10:23:27 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} y^{\prime }&=\frac {2 a \left (-y^{2}+4 a x -1\right )}{-y^{3}+4 a x y-y-2 a y^{6}+24 y^{4} a^{2} x -96 y^{2} a^{3} x^{2}+128 a^{4} x^{3}} \end{align*}

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 48

dsolve(diff(y(x),x) = 2*a*(-y(x)^2+4*a*x-1)/(-y(x)^3+4*y(x)*a*x-y(x)-2*a*y(x)^6+24*y(x)^4*a^2*x-96*y(x)^2*a^3*x^2+128*a^4*x^3),y(x), singsol=all)
 

\[ \frac {y}{2 a}-\frac {1}{16 a^{2} \left (-4 a x +y^{2}\right )^{2}}+\frac {1}{32 a^{3} x -8 a^{2} y^{2}}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.157 (sec). Leaf size: 381

DSolve[D[y[x],x] == (2*a*(-1 + 4*a*x - y[x]^2))/(128*a^4*x^3 - y[x] + 4*a*x*y[x] - 96*a^3*x^2*y[x]^2 - y[x]^3 + 24*a^2*x*y[x]^4 - 2*a*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,1\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,2\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,3\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,4\right ] \\ y(x)\to \text {Root}\left [8 \text {$\#$1}^5 a-16 \text {$\#$1}^4 a^2 c_1-64 \text {$\#$1}^3 a^2 x+\text {$\#$1}^2 \left (-2+128 a^3 c_1 x\right )+128 \text {$\#$1} a^3 x^2-256 a^4 c_1 x^2+8 a x-1\&,5\right ] \\ \end{align*}