Internal problem ID [8925]
Internal file name [OUTPUT/7860_Monday_June_06_2022_12_48_04_AM_63998398/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 591.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \sqrt {a}\, y-F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[-a/b/x, 1/y]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 126
dsolve(diff(y(x),x) = F((a*y(x)^2+b*x^2)/a)*x/a^(1/2)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (F \left (\frac {a \,\textit {\_Z}^{2}+b \,x^{2}}{a}\right ) \sqrt {a}+b \right ) \\ y \left (x \right ) &= \frac {\sqrt {a \left (-b \,x^{2}+\operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) \sqrt {a}+b}d \textit {\_a} \right ) b -b \,x^{2}+2 c_{1} a \right ) a \right )}}{a} \\ y \left (x \right ) &= -\frac {\sqrt {a \left (-b \,x^{2}+\operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) \sqrt {a}+b}d \textit {\_a} \right ) b -b \,x^{2}+2 c_{1} a \right ) a \right )}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.531 (sec). Leaf size: 253
DSolve[y'[x] == (x*F[(b*x^2 + a*y[x]^2)/a])/(Sqrt[a]*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {b K[2]}{b+\sqrt {a} F\left (\frac {b x^2+a K[2]^2}{a}\right )}-\int _1^x\left (\frac {2 b K[1] K[2] F'\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )}-\frac {2 b F\left (\frac {b K[1]^2+a K[2]^2}{a}\right ) K[1] K[2] F'\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {b F\left (\frac {b K[1]^2+a y(x)^2}{a}\right ) K[1]}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a y(x)^2}{a}\right )\right )}dK[1]=c_1,y(x)\right ] \]