problem 26

Internal problem ID [1132]
Internal ﬁle name [OUTPUT/1133_Sunday_June_05_2022_02_03_16_AM_33275409/index.tex]

Book: Elementary diﬀerential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number: 26.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "reduction_of_order", "second_order_change_of_variable_on_y_method_2"

\[ \boxed {4 x^{2} \sin \left (x \right ) y^{\prime \prime }-4 x \left (x \cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime }+\left (2 x \cos \left (x \right )+3 \sin \left (x \right )\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x} \end {align*}

Given one basis solution \(y_{1}\left (x \right )\), then the second basis solution is given by \[ y_{2}\left (x \right ) = y_{1} \left (\int \frac {{\mathrm e}^{-\left (\int p d x \right )}}{y_{1}^{2}}d x \right ) \] Where \(p(x)\) is the coeﬃcient of \(y^{\prime }\) when the ode is written in the normal form \[ y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y = f \left (x \right ) \] Looking at the ode to solve shows that \[ p \left (x \right ) = \frac {-4 x^{2} \cos \left (x \right )-4 \sin \left (x \right ) x}{4 \sin \left (x \right ) x^{2}} \] Therefore \begin{align*} y_{2}\left (x \right ) &= \sqrt {x}\, \left (\int \frac {{\mathrm e}^{-\left (\int \frac {-4 x^{2} \cos \left (x \right )-4 \sin \left (x \right ) x}{4 \sin \left (x \right ) x^{2}}d x \right )}}{x}d x \right ) \\ y_{2}\left (x \right ) &= \sqrt {x} \int \frac {{\mathrm e}^{\ln \left (\sin \left (x \right )\right )+\ln \left (x \right )}}{x} , dx \\ y_{2}\left (x \right ) &= \sqrt {x}\, \left (\int \sin \left (x \right )d x \right ) \\ y_{2}\left (x \right ) &= -\sqrt {x}\, \cos \left (x \right ) \\ \end{align*} Hence the solution is \begin{align*} y &= c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right ) \\ &= \sqrt {x}\, c_{1} -c_{2} \sqrt {x}\, \cos \left (x \right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {x}\, c_{1} -c_{2} \sqrt {x}\, \cos \left (x \right ) \\ \end{align*}

Veriﬁcation of solutions

\[ y = \sqrt {x}\, c_{1} -c_{2} \sqrt {x}\, \cos \left (x \right ) \] Veriﬁed OK.

Maple trace Kovacic algorithm successful

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
   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) 
   <- Kovacics algorithm successful 
   Change of variables used: 
      [x = arccos(t)] 
   Linear ODE actually solved: 
      (2*arccos(t)*t+3*(-t^2+1)^(1/2))*u(t)+(-4*arccos(t)*t^2+4*arccos(t))*diff(u(t),t)+(-4*arccos(t)^2*(-t^2+1)^(1/2)*t^2+4*arccos( 
<- change of variables successful`

\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} +c_{2} \cos \left (x \right )\right ) \]

9.26 problem 26