Internal problem ID [9217]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 883.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\[ \boxed {y^{\prime }-\frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+a \,x^{4} b^{2}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{\frac {7}{2}} y}=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 393
dsolve(diff(y(x),x) = (a^3+y(x)^4*a^3+2*y(x)^2*a^2*b*x^2+a*x^4*b^2+y(x)^6*a^3+3*y(x)^4*a^2*b*x^2+3*y(x)^2*a*b^2*x^4+b^3*x^6)*x/a^(7/2)/y(x),y(x), singsol=all)
\[ \frac {\int _{\textit {\_b}}^{x}\frac {\left (b^{3} \textit {\_a}^{6}+3 a \,b^{2} \textit {\_a}^{4} y \left (x \right )^{2}+3 a^{2} b \,\textit {\_a}^{2} y \left (x \right )^{4}+y \left (x \right )^{6} a^{3}+a \,b^{2} \textit {\_a}^{4}+2 a^{2} y \left (x \right )^{2} b \,\textit {\_a}^{2}+y \left (x \right )^{4} a^{3}+a^{3}\right ) \textit {\_a}}{y \left (x \right )^{6} a^{3}+3 a^{2} b \,\textit {\_a}^{2} y \left (x \right )^{4}+3 a \,b^{2} \textit {\_a}^{4} y \left (x \right )^{2}+b^{3} \textit {\_a}^{6}+y \left (x \right )^{4} a^{3}+2 a^{2} y \left (x \right )^{2} b \,\textit {\_a}^{2}+a \,b^{2} \textit {\_a}^{4}+a^{3}+a^{\frac {5}{2}} b}d \textit {\_a}}{a^{\frac {7}{2}}}-\left (\int _{}^{y \left (x \right )}\frac {2 \textit {\_f} \left (\frac {1}{2}+\left (a^{\frac {5}{2}} b +b^{3} x^{6}+3 a \,x^{4} \left (\textit {\_f}^{2}+\frac {1}{3}\right ) b^{2}+3 \left (\textit {\_f}^{2}+\frac {2}{3}\right ) a^{2} x^{2} \textit {\_f}^{2} b +\left (\textit {\_f}^{6}+\textit {\_f}^{4}+1\right ) a^{3}\right ) b \left (\int _{\textit {\_b}}^{x}\frac {\left (3 b \,\textit {\_a}^{2}+3 a \,\textit {\_f}^{2}+2 a \right ) \left (b \,\textit {\_a}^{2}+a \,\textit {\_f}^{2}\right ) \textit {\_a}}{\left (a^{\frac {5}{2}} b +\left (\textit {\_f}^{6}+\textit {\_f}^{4}+1\right ) a^{3}+3 \textit {\_a}^{2} \left (\textit {\_f}^{2}+\frac {2}{3}\right ) \textit {\_f}^{2} b \,a^{2}+3 \textit {\_a}^{4} b^{2} \left (\textit {\_f}^{2}+\frac {1}{3}\right ) a +b^{3} \textit {\_a}^{6}\right )^{2}}d \textit {\_a} \right )\right )}{a^{\frac {5}{2}} b +b^{3} x^{6}+3 a \,x^{4} \left (\textit {\_f}^{2}+\frac {1}{3}\right ) b^{2}+3 \left (\textit {\_f}^{2}+\frac {2}{3}\right ) a^{2} x^{2} \textit {\_f}^{2} b +\left (\textit {\_f}^{6}+\textit {\_f}^{4}+1\right ) a^{3}}d \textit {\_f} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.834 (sec). Leaf size: 164
DSolve[y'[x] == (x*(a^3 + a*b^2*x^4 + b^3*x^6 + 2*a^2*b*x^2*y[x]^2 + 3*a*b^2*x^4*y[x]^2 + a^3*y[x]^4 + 3*a^2*b*x^2*y[x]^4 + a^3*y[x]^6))/(a^(7/2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {x^2}{2}-\frac {1}{2} a^{5/2} \text {RootSum}\left [\text {$\#$1}^3 b^3+3 \text {$\#$1}^2 a b^2 y(x)^2+\text {$\#$1}^2 a b^2+3 \text {$\#$1} a^2 b y(x)^4+2 \text {$\#$1} a^2 b y(x)^2+a^{5/2} b+a^3 y(x)^6+a^3 y(x)^4+a^3\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 b^2+6 \text {$\#$1} a b y(x)^2+2 \text {$\#$1} a b+3 a^2 y(x)^4+2 a^2 y(x)^2}\&\right ]=c_1,y(x)\right ] \]