60.2.318 problem 895

Internal problem ID [10905]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 895
Date solved : Tuesday, January 28, 2025 at 05:30:26 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {\left (-256 a \,x^{2} y-32 a^{2} x^{6}-256 a \,x^{2}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 79

dsolve(diff(y(x),x) = (-256*a*x^2*y(x)-32*a^2*x^6-256*a*x^2+512*y(x)^3+192*x^4*a*y(x)^2+24*y(x)*a^2*x^8+a^3*x^12)*x/(512*y(x)+64*a*x^4+512),y(x), singsol=all)
 

\begin{align*} y &= \frac {8+\left (-\sqrt {-x^{2}+c_{1}}+1\right ) a \,x^{4}}{-8+8 \sqrt {-x^{2}+c_{1}}} \\ y &= \frac {-8+\left (-\sqrt {-x^{2}+c_{1}}-1\right ) a \,x^{4}}{8+8 \sqrt {-x^{2}+c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.483 (sec). Leaf size: 75

DSolve[D[y[x],x] == (x*(-256*a*x^2 - 32*a^2*x^6 + a^3*x^12 - 256*a*x^2*y[x] + 24*a^2*x^8*y[x] + 192*a*x^4*y[x]^2 + 512*y[x]^3))/(512 + 64*a*x^4 + 512*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {a x^4}{8}+\frac {512}{-512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8}-\frac {512}{512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8} \\ \end{align*}