12.14.7 problem 4

Internal problem ID [1948]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 4
Date solved : Monday, January 27, 2025 at 05:38:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+x \left (4 x^{2}+2 x +7\right ) y^{\prime }-\left (-7 x^{2}-4 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 48

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)+x*(7+2*x+4*x^2)*diff(y(x),x)-(1-4*x-7*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{5}/{4}} \left (1-\frac {1}{2} x -\frac {19}{104} x^{2}+\frac {1571}{10608} x^{3}+\frac {3225}{198016} x^{4}-\frac {752183}{29702400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+2 x -\frac {11}{6} x^{2}-\frac {1}{7} x^{3}+\frac {895}{1848} x^{4}-\frac {499}{13860} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 86

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]+x*(7+2*x+4*x^2)*D[y[x],x]-(1-4*x-7*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {752183 x^5}{29702400}+\frac {3225 x^4}{198016}+\frac {1571 x^3}{10608}-\frac {19 x^2}{104}-\frac {x}{2}+1\right )+\frac {c_2 \left (-\frac {499 x^5}{13860}+\frac {895 x^4}{1848}-\frac {x^3}{7}-\frac {11 x^2}{6}+2 x+1\right )}{x} \]