Internal problem ID [9190]
Internal file name [OUTPUT/8125_Monday_June_06_2022_01_50_36_AM_7168540/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 856.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {\left (-\frac {1}{x}-f_{1} \left (y^{2}-2 x \right )\right ) x}{\sqrt {y^{2}}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & f_{1} \left (y^{2}-2 x \right ) x -y^{\prime } \sqrt {y^{2}}+1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-f_{1} \left (y^{2}-2 x \right ) x -1}{\sqrt {y^{2}}} \end {array} \]
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 `, `-> Computing symmetries using: way = 5`[0, f__1(y^2-2*x)*((y^2)^(1/2)+y)/y^2]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 65
dsolve(diff(y(x),x) = -(-1/x-_F1(y(x)^2-2*x))/(y(x)^2)^(1/2)*x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {2 \operatorname {RootOf}\left (x^{2}-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (2 \textit {\_a} \right )}d \textit {\_a} \right )+4 c_{1} \right )+2 x} \\ y \left (x \right ) &= -\sqrt {2 \operatorname {RootOf}\left (x^{2}-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (2 \textit {\_a} \right )}d \textit {\_a} \right )+4 c_{1} \right )+2 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.33 (sec). Leaf size: 99
DSolve[y'[x] == (x*(x^(-1) + F1[-2*x + y[x]^2]))/Sqrt[y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {K[2]^2}}{\text {F1}\left (K[2]^2-2 x\right )}-\int _1^x\frac {2 K[2] \text {F1}'\left (K[2]^2-2 K[1]\right )}{\text {F1}\left (K[2]^2-2 K[1]\right )^2}dK[1]\right )dK[2]+\int _1^x\left (-K[1]-\frac {1}{\text {F1}\left (y(x)^2-2 K[1]\right )}\right )dK[1]=c_1,y(x)\right ] \]