Internal problem ID [12748]
Internal file name [OUTPUT/11401_Friday_November_03_2023_06_32_33_AM_59301967/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 4. N-th Order Linear Differential Equations. Exercises 4.1, page 186
Problem number: 6.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {\left (x^{2}-4\right ) y^{\prime \prime }+\ln \left (x \right ) y=x \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 2] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(x)y^{\prime } + q(x) y &= F \end {align*}
Where here \begin {align*} p(x) &=0\\ q(x) &=\frac {\ln \left (x \right )}{x^{2}-4}\\ F &=\frac {x \,{\mathrm e}^{x}}{x^{2}-4} \end {align*}
Hence the ode is \begin {align*} y^{\prime \prime }+\frac {\ln \left (x \right ) y}{x^{2}-4} = \frac {x \,{\mathrm e}^{x}}{x^{2}-4} \end {align*}
The domain of \(p(x)=0\) is \[
\{-\infty Maple trace
✗ Solution by Maple
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
Not solved
`Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
trying high order exact linear fully integrable
trying differential order: 2; linear nonhomogeneous with symmetry [0,1]
trying a double symmetry of the form [xi=0, eta=F(x)]
trying symmetries linear in x and y(x)
-> Try solving first the homogeneous part of the ODE
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
-> 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 a symmetry of the form [xi=0, eta=F(x)]
trying 2nd order exact linear
trying symmetries linear in x and y(x)
trying to convert to a linear ODE with constant coefficients
<- unable to find a useful change of variables
trying symmetries linear in x and y(x)
trying to convert to a linear ODE with constant coefficients
trying to convert to an ODE of Bessel type
-> trying reduction of order to Riccati
trying Riccati sub-methods:
trying Riccati_symmetries
-> trying a symmetry pattern of the form [F(x)*G(y), 0]
-> trying a symmetry pattern of the form [0, F(x)*G(y)]
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)]
--- Trying Lie symmetry methods, 2nd order ---
`, `-> Computing symmetries using: way = 3
`, `-> Computing symmetries using: way = 5`
dsolve([(x^2-4)*diff(y(x),x$2)+ln(x)*y(x)=x*exp(x),y(1) = 1, D(y)(1) = 2],y(x), singsol=all)
DSolve[{(x^2-4)*y''[x]+Log[x]*y[x]==x*Exp[x],{y[1]==1,y'[1]==2}},y[x],x,IncludeSingularSolutions -> True]