60.2.321 problem 898

Internal problem ID [10908]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 898
Date solved : Tuesday, January 28, 2025 at 05:30:45 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {32 x^{5} y+8 x^{3}+32 x^{5}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{16 x^{6} \left (4 x^{2} y+1+4 x^{2}\right )} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 87

dsolve(diff(y(x),x) = 1/16/x^6*(32*x^5*y(x)+8*x^3+32*x^5+64*x^6*y(x)^3+48*x^4*y(x)^2+12*x^2*y(x)+1)/(4*x^2*y(x)+1+4*x^2),y(x), singsol=all)
 

\begin{align*} y &= \frac {4 x^{2}-\sqrt {\frac {c_{1} x +2}{x}}+1}{4 \left (\sqrt {\frac {c_{1} x +2}{x}}-1\right ) x^{2}} \\ y &= \frac {-4 x^{2}-\sqrt {\frac {c_{1} x +2}{x}}-1}{4 \left (\sqrt {\frac {c_{1} x +2}{x}}+1\right ) x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.814 (sec). Leaf size: 106

DSolve[D[y[x],x] == (1/16 + x^3/2 + 2*x^5 + (3*x^2*y[x])/4 + 2*x^5*y[x] + 3*x^4*y[x]^2 + 4*x^6*y[x]^3)/(x^6*(1 + 4*x^2 + 4*x^2*y[x])),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {256 x^2-\sqrt {\frac {8192}{x}+c_1}+64}{4 x^2 \left (-64+\sqrt {\frac {8192}{x}+c_1}\right )} \\ y(x)\to -\frac {256 x^2+\sqrt {\frac {8192}{x}+c_1}+64}{4 x^2 \left (64+\sqrt {\frac {8192}{x}+c_1}\right )} \\ y(x)\to -\frac {1}{4 x^2} \\ \end{align*}