54.7.11 problem 11

Internal problem ID [8659]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 11
Date solved : Monday, January 27, 2025 at 04:18:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-5 x y^{\prime }+\left (8+5 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 74

Order:=8; 
dsolve(x^2*diff(y(x),x$2)-5*x*diff(y(x),x)+(8+5*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{2} \left (c_{1} x^{2} \left (1-\frac {5}{3} x +\frac {25}{24} x^{2}-\frac {25}{72} x^{3}+\frac {125}{1728} x^{4}-\frac {125}{12096} x^{5}+\frac {625}{580608} x^{6}-\frac {3125}{36578304} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (25 x^{2}-\frac {125}{3} x^{3}+\frac {625}{24} x^{4}-\frac {625}{72} x^{5}+\frac {3125}{1728} x^{6}-\frac {3125}{12096} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-10 x +\frac {500}{9} x^{3}-\frac {15625}{288} x^{4}+\frac {19625}{864} x^{5}-\frac {56875}{10368} x^{6}+\frac {443125}{508032} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 123

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-5*x*D[y[x],x]+(8+5*x)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^2 \left (33125 x^6-140250 x^5+348750 x^4-396000 x^3+64800 x^2+51840 x+10368\right )}{10368}-\frac {25 x^4 \left (125 x^4-600 x^3+1800 x^2-2880 x+1728\right ) \log (x)}{3456}\right )+c_2 \left (\frac {625 x^{10}}{580608}-\frac {125 x^9}{12096}+\frac {125 x^8}{1728}-\frac {25 x^7}{72}+\frac {25 x^6}{24}-\frac {5 x^5}{3}+x^4\right ) \]