Internal problem ID [8321]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 741.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (a y^{2}+x^{2} b \right )^{3} x}{a^{\frac {5}{2}} \left (a y^{2}+x^{2} b +a \right ) y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 400
dsolve(diff(y(x),x) = (a*y(x)^2+b*x^2)^3/a^(5/2)*x/(a*y(x)^2+b*x^2+a)/y(x),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{x}\frac {\left (b \,\textit {\_a}^{2}+a y \relax (x )^{2}\right )^{3} \textit {\_a}}{a^{3} \left (y \relax (x )^{6} a^{3}+3 a^{2} b \,\textit {\_a}^{2} y \relax (x )^{4}+3 a \,b^{2} \textit {\_a}^{4} y \relax (x )^{2}+b^{3} \textit {\_a}^{6}+a^{\frac {5}{2}} b y \relax (x )^{2}+a^{\frac {3}{2}} b^{2} \textit {\_a}^{2}+a^{\frac {5}{2}} b \right )}d \textit {\_a} +\int _{}^{y \relax (x )}\left (-\frac {\left (a \,\textit {\_f}^{2}+b \,x^{2}+a \right ) \textit {\_f}}{\sqrt {a}\, \left (a^{3} \textit {\_f}^{6}+3 a^{2} b \,x^{2} \textit {\_f}^{4}+3 a \,b^{2} x^{4} \textit {\_f}^{2}+b^{3} x^{6}+a^{\frac {5}{2}} b \,\textit {\_f}^{2}+a^{\frac {3}{2}} b^{2} x^{2}+a^{\frac {5}{2}} b \right )}-\left (\int _{\textit {\_b}}^{x}\left (-\frac {\left (b \,\textit {\_a}^{2}+a \,\textit {\_f}^{2}\right )^{3} \textit {\_a} \left (6 a^{3} \textit {\_f}^{5}+12 a^{2} b \,\textit {\_a}^{2} \textit {\_f}^{3}+6 a \,b^{2} \textit {\_a}^{4} \textit {\_f} +2 b \textit {\_f} \,a^{\frac {5}{2}}\right )}{a^{3} \left (a^{3} \textit {\_f}^{6}+3 a^{2} b \,\textit {\_a}^{2} \textit {\_f}^{4}+3 a \,b^{2} \textit {\_a}^{4} \textit {\_f}^{2}+b^{3} \textit {\_a}^{6}+a^{\frac {5}{2}} b \,\textit {\_f}^{2}+a^{\frac {3}{2}} b^{2} \textit {\_a}^{2}+a^{\frac {5}{2}} b \right )^{2}}+\frac {6 \left (b \,\textit {\_a}^{2}+a \,\textit {\_f}^{2}\right )^{2} \textit {\_a} \textit {\_f}}{a^{2} \left (a^{3} \textit {\_f}^{6}+3 a^{2} b \,\textit {\_a}^{2} \textit {\_f}^{4}+3 a \,b^{2} \textit {\_a}^{4} \textit {\_f}^{2}+b^{3} \textit {\_a}^{6}+a^{\frac {5}{2}} b \,\textit {\_f}^{2}+a^{\frac {3}{2}} b^{2} \textit {\_a}^{2}+a^{\frac {5}{2}} b \right )}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.816 (sec). Leaf size: 175
DSolve[y'[x] == (x*(b*x^2 + a*y[x]^2)^3)/(a^(5/2)*y[x]*(a + b*x^2 + a*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (x^2-a^{3/2} \text {RootSum}\left [\text {$\#$1}^3 b^3+3 \text {$\#$1}^2 a b^2 y(x)^2+\text {$\#$1} a^{3/2} b^2+3 \text {$\#$1} a^2 b y(x)^4+a^{5/2} b y(x)^2+a^{5/2} b+a^3 y(x)^6\&,\frac {a y(x)^2 \log \left (x^2-\text {$\#$1}\right )+a \log \left (x^2-\text {$\#$1}\right )+\text {$\#$1} b \log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 b^2+6 \text {$\#$1} a b y(x)^2+a^{3/2} b+3 a^2 y(x)^4}\&\right ]\right )=c_1,y(x)\right ] \]