Internal problem ID [5473]
Internal file name [OUTPUT/4721_Sunday_June_05_2022_03_04_02_PM_75632679/index.tex]
Book: A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd.
New York, reprinted 1956
Section: Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius.
page 243
Problem number: Ex. 6(ii), page 257.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 3; missing the dependent variable trying Louvillian solutions for 3rd order ODEs, imprimitive case Louvillian solutions for 3rd order ODEs, imprimitive case: input is reducible, switching to DFactorsols checking if the LODE is of Euler type expon. solutions partially successful. Result(s) =`, [x^2]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 79
Order:=6; dsolve(x^3*(1+x)*diff(y(x),x$3)-(2+4*x)*x^2*diff(y(x),x$2)+(4+10*x)*x*diff(y(x),x)-(4+12*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = x \left (\left (2 x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )^{2} c_{3} +\ln \left (x \right ) \left (2+\operatorname {O}\left (x^{6}\right )\right ) c_{2} x +2 \left (\left (-4\right ) x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{3} +\left (5+\operatorname {O}\left (x^{6}\right )\right ) c_{2} x +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (2+4 x +2 x^{2}+\operatorname {O}\left (x^{6}\right )\right ) c_{3} \right ) \]
✓ Solution by Mathematica
Time used: 0.514 (sec). Leaf size: 49
AsymptoticDSolveValue[x^3*(1+x)*y'''[x]-(2+4*x)*x^2*y''[x]+(4+10*x)*x*y'[x]-(4+12*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 x^2+c_1 \left (2 \left (x^2+11 x+1\right ) x+2 x^2 \log ^2(x)-14 x^2 \log (x)\right )+c_3 x^2 \log (x) \]