1.114 problem 114

Internal problem ID [9096]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 114
Date solved : Thursday, October 17, 2024 at 01:19:27 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solve

\begin{align*} x y^{\prime }-x \sqrt {y^{2}+x^{2}}-y&=0 \end{align*}

Unknown ode type.

1.114.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (\frac {d}{d x}y \left (x \right )\right )-x \sqrt {y \left (x \right )^{2}+x^{2}}-y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {x \sqrt {y \left (x \right )^{2}+x^{2}}+y \left (x \right )}{x} \end {array} \]

1.114.2 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 
-> Calling odsolve with the ODE`, diff(diff(y(x), x), x)-(x^2*y(x)+2*x*(diff(y(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)]
 

1.114.3 Maple dsolve solution

Solving time : 3.720 (sec)
Leaf size : 28

dsolve(x*diff(y(x),x)-x*(x^2+y(x)^2)^(1/2)-y(x) = 0, 
       y(x),singsol=all)
 

\[ \ln \left (\sqrt {x^{2}+y^{2}}+y\right )-x -\ln \left (x \right )-c_{1} = 0 \]

1.114.4 Mathematica DSolve solution

Solving time : 31.54 (sec)
Leaf size : 46

DSolve[{x*D[y[x],x] - x*Sqrt[y[x]^2 + x^2] - y[x]==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {x \tanh (x+c_1)}{\sqrt {\text {sech}^2(x+c_1)}} \\ y(x)\to \frac {x \tanh (x+c_1)}{\sqrt {\text {sech}^2(x+c_1)}} \\ \end{align*}