Internal problem ID [2400]
Internal file name [OUTPUT/2400_Tuesday_February_27_2024_08_36_55_AM_96829618/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 41, page 195
Problem number: 25.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{3} \left (x^{2}+3\right ) y^{\prime \prime }+5 y^{\prime } x -y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ \left (x^{5}+3 x^{3}\right ) y^{\prime \prime }+5 y^{\prime } x +\left (-x -1\right ) y = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= \frac {5}{x^{2} \left (x^{2}+3\right )}\\ q(x) &= -\frac {x +1}{x^{3} \left (x^{2}+3\right )}\\ \end {align*}
| \(p(x)=\frac {5}{x^{2} \left (x^{2}+3\right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``irregular''}\) |
| \(x = -i \sqrt {3}\) | \(\text {``regular''}\) |
| \(x = i \sqrt {3}\) | \(\text {``regular''}\) |
| \(q(x)=-\frac {x +1}{x^{3} \left (x^{2}+3\right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``irregular''}\) |
| \(x = -i \sqrt {3}\) | \(\text {``regular''}\) |
| \(x = i \sqrt {3}\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \(\left [-i \sqrt {3}, i \sqrt {3}, \infty \right ]\)
Irregular singular points : \([0]\)
Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating.
Verification of solutions N/A
Maple trace
`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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunC ODE, case a <> 0, e <> 0, c <> 0 `
✗ Solution by Maple
Order:=6; dsolve(x^3*(3+x^2)*diff(y(x),x$2)+5*x*diff(y(x),x)-(1+x)*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.053 (sec). Leaf size: 99
AsymptoticDSolveValue[x^3*(3+x^2)*y''[x]+5*x*y'[x]-(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {18968303719 x^5}{1220703125000}-\frac {20383193 x^4}{1953125000}+\frac {26731 x^3}{3906250}+\frac {259 x^2}{31250}+\frac {37 x}{125}+1\right ) \sqrt [5]{x}+c_2 e^{\left .\frac {5}{3}\right /x} \left (\frac {869909160612721304 x^5}{27030487060546875}+\frac {46847788879262 x^4}{4805419921875}+\frac {15542572604 x^3}{4271484375}+\frac {2270672 x^2}{1265625}+\frac {1372 x}{1125}+1\right ) x^{9/5} \]