41.1.6 problem Ex. 6(v), page 257

Internal problem ID [6819]
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(v), page 257
Date solved : Monday, January 27, 2025 at 02:31:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-x \right ) x^{2} y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 38

Order:=6; 
dsolve((1-x)*x^2*diff(y(x),x$2)+(5*x-4)*x*diff(y(x),x)+(6-9*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{2} \left (\ln \left (x \right ) \left (x +\operatorname {O}\left (x^{6}\right )\right ) c_2 +c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x +\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.055 (sec). Leaf size: 30

AsymptoticDSolveValue[(1-x)*x^2*D[y[x],{x,2}]+(5*x-4)*x*D[y[x],x]+(6-9*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 x^3+c_1 \left (x^3 \log (x)-x^2 (3 x-1)\right ) \]