Internal problem ID [4723]
Internal file name [OUTPUT/4216_Sunday_June_05_2022_12_42_30_PM_62461942/index.tex
]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XV. page 194
Problem number: 13.
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, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x}=\sqrt {x}} \] 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. \[ 2 x^{2} y^{\prime \prime }+\left (-3 x -2\right ) y^{\prime }+\left (2-\frac {1}{x}\right ) y = 0 \] 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 {3 x +2}{2 x^{2}}\\ q(x) &= \frac {2 x -1}{2 x^{3}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([\infty ]\)
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 Kovacic algorithm successful
`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)] -> Try solving first the homogeneous part of the ODE 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) Group is reducible, not completely reducible Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre <- Kummer successful <- special function solution successful Solution using Kummer functions still has integrals. Trying a hypergeometric solution. -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE -> Trying to convert hypergeometric functions to elementary form... <- elementary form for at least one hypergeometric solution is achieved - returning with no uncomputed integrals <- Kovacics algorithm successful <- solving first the homogeneous part of the ODE successful`
✗ Solution by Maple
Order:=6; dsolve(2*x^2*diff(y(x),x$2)-(3*x+2)*diff(y(x),x)+(2*x-1)/x*y(x)=x^(1/2),y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.158 (sec). Leaf size: 222
AsymptoticDSolveValue[2*x^2*y''[x]-(3*x+2)*y'[x]+(2*x-1)/x*y[x]==x^(1/2),y[x],{x,0,5}]
\[ y(x)\to \frac {1}{256} e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^4 \left (\frac {2 e^{\frac {1}{x}} \left (15663375 x^7+20072325 x^6+10329540 x^5+4131816 x^4+2754544 x^3+5509088 x^2-64 x-64\right )}{x^{3/2}}-11018112 \sqrt {\pi } \text {erfi}\left (\frac {1}{\sqrt {x}}\right )\right )+c_2 e^{-1/x} \left (-\frac {405405 x^5}{16}+\frac {45045 x^4}{16}-\frac {693 x^3}{2}+\frac {189 x^2}{4}-7 x+1\right ) x^4+\frac {\left (\frac {5 x}{2}+1\right ) \left (-\frac {15015 x^6}{64}+\frac {693 x^5}{20}-\frac {189 x^4}{32}+\frac {7 x^3}{6}-\frac {x^2}{4}\right )}{\sqrt {x}}+\frac {c_1 \left (\frac {5 x}{2}+1\right )}{\sqrt {x}} \]