Internal problem ID [8463]
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)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+b^{2} x^{4} a +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} \end {gather*}
✓ Solution by Maple
Time used: 0.151 (sec). Leaf size: 595
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)
\[ \int _{\textit {\_b}}^{x}\frac {\left (\textit {\_a}^{6} b^{3}+3 \textit {\_a}^{4} y \relax (x )^{2} b^{2} a +3 y \relax (x )^{4} a^{2} b \,\textit {\_a}^{2}+y \relax (x )^{6} a^{3}+a \,b^{2} \textit {\_a}^{4}+2 a^{2} y \relax (x )^{2} b \,\textit {\_a}^{2}+y \relax (x )^{4} a^{3}+a^{3}\right ) \textit {\_a}}{\left (y \relax (x )^{6} a^{3}+3 y \relax (x )^{4} a^{2} b \,\textit {\_a}^{2}+3 \textit {\_a}^{4} y \relax (x )^{2} b^{2} a +\textit {\_a}^{6} b^{3}+y \relax (x )^{4} a^{3}+2 a^{2} y \relax (x )^{2} b \,\textit {\_a}^{2}+a \,b^{2} \textit {\_a}^{4}+a^{3}+a^{\frac {5}{2}} b \right ) a^{\frac {7}{2}}}d \textit {\_a} +\int _{}^{y \relax (x )}\left (-\frac {\textit {\_f}}{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^{3} \textit {\_f}^{4}+2 a^{2} \textit {\_f}^{2} b \,x^{2}+a \,b^{2} x^{4}+a^{3}+a^{\frac {5}{2}} b}-\left (\int _{\textit {\_b}}^{x}\left (\frac {\left (6 a \,b^{2} \textit {\_a}^{4} \textit {\_f} +12 a^{2} b \,\textit {\_a}^{2} \textit {\_f}^{3}+6 a^{3} \textit {\_f}^{5}+4 \textit {\_a}^{2} \textit {\_f} \,a^{2} b +4 \textit {\_f}^{3} a^{3}\right ) \textit {\_a}}{\left (a^{3} \textit {\_f}^{6}+3 \textit {\_a}^{2} \textit {\_f}^{4} a^{2} b +3 \textit {\_a}^{4} \textit {\_f}^{2} a \,b^{2}+\textit {\_a}^{6} b^{3}+a^{3} \textit {\_f}^{4}+2 a^{2} \textit {\_f}^{2} b \,\textit {\_a}^{2}+a \,b^{2} \textit {\_a}^{4}+a^{3}+a^{\frac {5}{2}} b \right ) a^{\frac {7}{2}}}-\frac {\left (\textit {\_a}^{6} b^{3}+3 \textit {\_a}^{4} \textit {\_f}^{2} a \,b^{2}+3 \textit {\_a}^{2} \textit {\_f}^{4} a^{2} b +a^{3} \textit {\_f}^{6}+a \,b^{2} \textit {\_a}^{4}+2 a^{2} \textit {\_f}^{2} b \,\textit {\_a}^{2}+a^{3} \textit {\_f}^{4}+a^{3}\right ) \textit {\_a} \left (6 a \,b^{2} \textit {\_a}^{4} \textit {\_f} +12 a^{2} b \,\textit {\_a}^{2} \textit {\_f}^{3}+6 a^{3} \textit {\_f}^{5}+4 \textit {\_a}^{2} \textit {\_f} \,a^{2} b +4 \textit {\_f}^{3} a^{3}\right )}{\left (a^{3} \textit {\_f}^{6}+3 \textit {\_a}^{2} \textit {\_f}^{4} a^{2} b +3 \textit {\_a}^{4} \textit {\_f}^{2} a \,b^{2}+\textit {\_a}^{6} b^{3}+a^{3} \textit {\_f}^{4}+2 a^{2} \textit {\_f}^{2} b \,\textit {\_a}^{2}+a \,b^{2} \textit {\_a}^{4}+a^{3}+a^{\frac {5}{2}} b \right )^{2} a^{\frac {7}{2}}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.801 (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]
\[ \text {Solve}\left [\frac {x^2}{2}-\frac {1}{2} a^{5/2} \text {RootSum}\left [\text {$\#$1}^3 b^3+3 \text {$\#$1}^2 a b^2 y(x)^2+\text {$\#$1}^2 a b^2+3 \text {$\#$1} a^2 b y(x)^4+2 \text {$\#$1} a^2 b y(x)^2+a^{5/2} b+a^3 y(x)^6+a^3 y(x)^4+a^3\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 b^2+6 \text {$\#$1} a b y(x)^2+2 \text {$\#$1} a b+3 a^2 y(x)^4+2 a^2 y(x)^2}\&\right ]=c_1,y(x)\right ] \]