Internal problem ID [3452]
Internal file name [OUTPUT/2945_Sunday_June_05_2022_08_47_44_AM_68847625/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 7
Problem number: 196.
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.
\[ \boxed {x y^{\prime }-y-\sqrt {x^{2}+y^{2}}\, x=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-y-\sqrt {x^{2}+y^{2}}\, x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y+\sqrt {x^{2}+y^{2}}\, x}{x} \end {array} \]
Maple trace Kovacic algorithm successful
`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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x)-(x^2*y(x)+2*(diff(y(x), x))*x-2*y(x))/x^2, y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Reducible group (found another exponential solution) <- Kovacics algorithm successful --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5`[0, (x^2+y^2)^(1/2)]
✓ Solution by Maple
Time used: 0.813 (sec). Leaf size: 28
dsolve(x*diff(y(x),x) = y(x)+x*sqrt(x^2+y(x)^2),y(x), singsol=all)
\[ \ln \left (y \left (x \right )+\sqrt {x^{2}+y \left (x \right )^{2}}\right )-x -\ln \left (x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.282 (sec). Leaf size: 30
DSolve[x y'[x]==y[x]+x Sqrt[x^2+y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} x e^{-x-c_1} \left (-1+e^{2 (x+c_1)}\right ) \]