Internal problem ID [8458]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 878.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 444
dsolve(diff(y(x),x) = (1+y(x)^4-8*a*x*y(x)^2+16*a^2*x^2+y(x)^6-12*y(x)^4*a*x+48*y(x)^2*a^2*x^2-64*a^3*x^3)/y(x),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{x}\frac {64 \textit {\_a}^{3} a^{3}-48 \textit {\_a}^{2} a^{2} y \relax (x )^{2}+12 \textit {\_a} a y \relax (x )^{4}-y \relax (x )^{6}-16 \textit {\_a}^{2} a^{2}+8 \textit {\_a} a y \relax (x )^{2}-y \relax (x )^{4}-1}{64 \textit {\_a}^{3} a^{3}-48 \textit {\_a}^{2} a^{2} y \relax (x )^{2}+12 \textit {\_a} a y \relax (x )^{4}-y \relax (x )^{6}-16 \textit {\_a}^{2} a^{2}+8 \textit {\_a} a y \relax (x )^{2}-y \relax (x )^{4}+2 a -1}d \textit {\_a} +\int _{}^{y \relax (x )}\left (\frac {\textit {\_f}}{-\textit {\_f}^{6}+12 \textit {\_f}^{4} a x -48 \textit {\_f}^{2} a^{2} x^{2}+64 a^{3} x^{3}-\textit {\_f}^{4}+8 \textit {\_f}^{2} a x -16 x^{2} a^{2}+2 a -1}-\left (\int _{\textit {\_b}}^{x}\left (\frac {-96 \textit {\_a}^{2} \textit {\_f} \,a^{2}+48 \textit {\_a} \,\textit {\_f}^{3} a -6 \textit {\_f}^{5}+16 \textit {\_a} \textit {\_f} a -4 \textit {\_f}^{3}}{64 \textit {\_a}^{3} a^{3}-48 \textit {\_a}^{2} \textit {\_f}^{2} a^{2}+12 \textit {\_a} \,\textit {\_f}^{4} a -\textit {\_f}^{6}-16 \textit {\_a}^{2} a^{2}+8 \textit {\_a} \,\textit {\_f}^{2} a -\textit {\_f}^{4}+2 a -1}-\frac {\left (64 \textit {\_a}^{3} a^{3}-48 \textit {\_a}^{2} \textit {\_f}^{2} a^{2}+12 \textit {\_a} \,\textit {\_f}^{4} a -\textit {\_f}^{6}-16 \textit {\_a}^{2} a^{2}+8 \textit {\_a} \,\textit {\_f}^{2} a -\textit {\_f}^{4}-1\right ) \left (-96 \textit {\_a}^{2} \textit {\_f} \,a^{2}+48 \textit {\_a} \,\textit {\_f}^{3} a -6 \textit {\_f}^{5}+16 \textit {\_a} \textit {\_f} a -4 \textit {\_f}^{3}\right )}{\left (64 \textit {\_a}^{3} a^{3}-48 \textit {\_a}^{2} \textit {\_f}^{2} a^{2}+12 \textit {\_a} \,\textit {\_f}^{4} a -\textit {\_f}^{6}-16 \textit {\_a}^{2} a^{2}+8 \textit {\_a} \,\textit {\_f}^{2} a -\textit {\_f}^{4}+2 a -1\right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.248 (sec). Leaf size: 130
DSolve[y'[x] == (1 + 16*a^2*x^2 - 64*a^3*x^3 - 8*a*x*y[x]^2 + 48*a^2*x^2*y[x]^2 + y[x]^4 - 12*a*x*y[x]^4 + y[x]^6)/y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [2 a \left (x-\frac {1}{2} \text {RootSum}\left [64 \text {$\#$1}^3 a^3-48 \text {$\#$1}^2 a^2 y(x)^2-16 \text {$\#$1}^2 a^2+12 \text {$\#$1} a y(x)^4+8 \text {$\#$1} a y(x)^2+2 a-y(x)^6-y(x)^4-1\&,\frac {\log (x-\text {$\#$1})}{48 \text {$\#$1}^2 a^2-24 \text {$\#$1} a y(x)^2-8 \text {$\#$1} a+3 y(x)^4+2 y(x)^2}\&\right ]\right )=c_1,y(x)\right ] \]