41.1.5 problem Ex. 6(iv), page 257

Internal problem ID [6818]
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(iv), page 257
Date solved : Monday, January 27, 2025 at 02:31:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 \left (2-x \right ) x^{2} y^{\prime \prime }-\left (4-x \right ) x y^{\prime }+\left (3-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 46

Order:=6; 
dsolve(2*(2-x)*x^2*diff(y(x),x$2)-(4-x)*x*diff(y(x),x)+(3-x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (x \left (1+\frac {1}{8} x +\frac {1}{32} x^{2}+\frac {5}{512} x^{3}+\frac {7}{2048} x^{4}+\frac {21}{16384} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_1 +\left (1+\frac {1}{4} x +\frac {1}{32} x^{2}+\frac {1}{128} x^{3}+\frac {5}{2048} x^{4}+\frac {7}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.057 (sec). Leaf size: 94

AsymptoticDSolveValue[2*(2-x)*x^2*D[y[x],{x,2}]-(4-x)*x*D[y[x],x]+(3-x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {5 x^{9/2}}{2048}-\frac {x^{7/2}}{128}-\frac {x^{5/2}}{32}-\frac {x^{3/2}}{4}+\sqrt {x}\right )+c_2 \left (\frac {7 x^{11/2}}{2048}+\frac {5 x^{9/2}}{512}+\frac {x^{7/2}}{32}+\frac {x^{5/2}}{8}+x^{3/2}\right ) \]