Internal problem ID [9075]
Internal file name [OUTPUT/8010_Monday_June_06_2022_01_15_20_AM_47718434/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (a y^{2}+x^{2} b \right )^{3} x}{a^{{5}/{2}} \left (a y^{2}+x^{2} b +a \right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{3} a^{{7}/{2}}+y^{\prime } y a^{{5}/{2}} b \,x^{2}-y^{6} a^{3} x -3 y^{4} a^{2} b \,x^{3}-3 y^{2} a \,b^{2} x^{5}-b^{3} x^{7}+y^{\prime } y a^{{7}/{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{6} a^{3} x +3 y^{4} a^{2} b \,x^{3}+3 y^{2} a \,b^{2} x^{5}+b^{3} x^{7}}{a^{{7}/{2}} y^{3}+a^{{5}/{2}} y b \,x^{2}+a^{{7}/{2}} y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 2`[0, (y^6*a^(7/2)+3*y^4*x^2*a^(5/2)*b+3*y^2*x^4*b^2*a^(3/2)+x^6*b^3*a^(1/2)+b*a^
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 242
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)
\[ \frac {\int _{\textit {\_b}}^{x}\frac {\left (b \,\textit {\_a}^{2}+a y \left (x \right )^{2}\right )^{3} \textit {\_a}}{b \left (y \left (x \right )^{2}+1\right ) a^{\frac {5}{2}}+a^{\frac {3}{2}} b^{2} \textit {\_a}^{2}+\left (b \,\textit {\_a}^{2}+a y \left (x \right )^{2}\right )^{3}}d \textit {\_a}}{a^{3}}-\frac {\int _{}^{y \left (x \right )}\frac {2 \left (\left (b \left (\textit {\_f}^{2}+1\right ) a^{\frac {5}{2}}+a^{\frac {3}{2}} b^{2} x^{2}+\left (a \,\textit {\_f}^{2}+b \,x^{2}\right )^{3}\right ) b \left (\int _{\textit {\_b}}^{x}\frac {\left (b \,\textit {\_a}^{2}+a \,\textit {\_f}^{2}\right )^{2} \textit {\_a} \left (2 b \,\textit {\_a}^{2}+2 a \,\textit {\_f}^{2}+3 a \right )}{{\left (b \left (\textit {\_f}^{2}+1\right ) a^{\frac {5}{2}}+a^{\frac {3}{2}} b^{2} \textit {\_a}^{2}+\left (b \,\textit {\_a}^{2}+a \,\textit {\_f}^{2}\right )^{3}\right )}^{2}}d \textit {\_a} \right )+\frac {b \,x^{2}}{2}+\frac {a \left (\textit {\_f}^{2}+1\right )}{2}\right ) \textit {\_f}}{b \left (\textit {\_f}^{2}+1\right ) a^{\frac {5}{2}}+a^{\frac {3}{2}} b^{2} x^{2}+\left (a \,\textit {\_f}^{2}+b \,x^{2}\right )^{3}}d \textit {\_f}}{\sqrt {a}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.84 (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]
\[ \text {Solve}\left [\frac {1}{2} \left (x^2-a^{3/2} \text {RootSum}\left [\text {$\#$1}^3 b^3+3 \text {$\#$1}^2 a b^2 y(x)^2+\text {$\#$1} a^{3/2} b^2+3 \text {$\#$1} a^2 b y(x)^4+a^{5/2} b y(x)^2+a^{5/2} b+a^3 y(x)^6\&,\frac {a y(x)^2 \log \left (x^2-\text {$\#$1}\right )+a \log \left (x^2-\text {$\#$1}\right )+\text {$\#$1} b \log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 b^2+6 \text {$\#$1} a b y(x)^2+a^{3/2} b+3 a^2 y(x)^4}\&\right ]\right )=c_1,y(x)\right ] \]