Internal problem ID [9285]
Internal file name [OUTPUT/8221_Monday_June_06_2022_02_20_31_AM_50054977/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 952.
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(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {-y+x^{2} \sqrt {x^{2}+y^{2}}-x \sqrt {x^{2}+y^{2}}\, y+x^{4} \sqrt {x^{2}+y^{2}}-y x^{3} \sqrt {x^{2}+y^{2}}+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \sqrt {x^{2}+y^{2}}\, y-x^{5} \sqrt {x^{2}+y^{2}}+y x^{3} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}+x \sqrt {x^{2}+y^{2}}\, y-x^{2} \sqrt {x^{2}+y^{2}}-y^{\prime } x +y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y+x^{2} \sqrt {x^{2}+y^{2}}-x \sqrt {x^{2}+y^{2}}\, y+x^{4} \sqrt {x^{2}+y^{2}}-y x^{3} \sqrt {x^{2}+y^{2}}+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x} \end {array} \]
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 = 5`[0, (x^2+y^2)^(1/2)*(x-y)/x]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 63
dsolve(diff(y(x),x) = -(-y(x)+(y(x)^2+x^2)^(1/2)*x^2-x*(y(x)^2+x^2)^(1/2)*y(x)+x^4*(y(x)^2+x^2)^(1/2)-x^3*(y(x)^2+x^2)^(1/2)*y(x)+x^5*(y(x)^2+x^2)^(1/2)-x^4*(y(x)^2+x^2)^(1/2)*y(x))/x,y(x), singsol=all)
\[ \ln \left (\frac {x \left (\sqrt {2 y \left (x \right )^{2}+2 x^{2}}+y \left (x \right )+x \right )}{y \left (x \right )-x}\right )+\frac {\left (4 x^{5}+5 x^{4}+10 x^{2}\right ) \sqrt {2}}{20}-c_{1} +\ln \left (2\right )-\ln \left (x \right ) = 0 \]
✓ Solution by Mathematica
Time used: 2.062 (sec). Leaf size: 120
DSolve[y'[x] == (y[x] - x^2*Sqrt[x^2 + y[x]^2] - x^4*Sqrt[x^2 + y[x]^2] - x^5*Sqrt[x^2 + y[x]^2] + x*y[x]*Sqrt[x^2 + y[x]^2] + x^3*y[x]*Sqrt[x^2 + y[x]^2] + x^4*y[x]*Sqrt[x^2 + y[x]^2])/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \tanh \left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{20 \sqrt {2}}\right ) \left (2+\sqrt {2} \tanh \left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{20 \sqrt {2}}\right )\right )}{\sqrt {2}+2 \tanh \left (\frac {4 x^5+5 x^4+10 x^2+20 c_1}{20 \sqrt {2}}\right )} \\ y(x)\to x \\ \end{align*}