49.19.2 problem 1(ii)

Internal problem ID [7722]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 166
Problem number : 1(ii)
Date solved : Monday, January 27, 2025 at 03:10:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 76

Order:=8; 
dsolve(4*x^2*diff(y(x),x$2)-4*x*exp(x)*diff(y(x),x)+3*cos(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (x \left (1+\frac {3}{4} x +\frac {1}{2} x^{2}+\frac {103}{384} x^{3}+\frac {669}{5120} x^{4}+\frac {54731}{921600} x^{5}+\frac {123443}{4838400} x^{6}+\frac {30273113}{2890137600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{1} +c_{2} \left (\ln \left (x \right ) \left (\frac {1}{2} x +\frac {3}{8} x^{2}+\frac {1}{4} x^{3}+\frac {103}{768} x^{4}+\frac {669}{10240} x^{5}+\frac {54731}{1843200} x^{6}+\frac {123443}{9676800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1+x +\frac {3}{4} x^{2}+\frac {59}{144} x^{3}+\frac {5701}{27648} x^{4}+\frac {17519}{184320} x^{5}+\frac {6852157}{165888000} x^{6}+\frac {417496453}{24385536000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.133 (sec). Leaf size: 146

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]-4*x*Exp[x]*D[y[x],x]+3*Cos[x]*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {123443 x^{15/2}}{4838400}+\frac {54731 x^{13/2}}{921600}+\frac {669 x^{11/2}}{5120}+\frac {103 x^{9/2}}{384}+\frac {x^{7/2}}{2}+\frac {3 x^{5/2}}{4}+x^{3/2}\right )+c_1 \left (\frac {\left (54731 x^5+120420 x^4+247200 x^3+460800 x^2+691200 x+921600\right ) x^{3/2} \log (x)}{1843200}+\frac {\left (1926367 x^6+4929300 x^5+11958000 x^4+26496000 x^3+62208000 x^2+82944000 x+165888000\right ) \sqrt {x}}{165888000}\right ) \]