54.4.4 problem 4

Internal problem ID [8588]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 4
Date solved : Monday, January 27, 2025 at 04:16:53 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x y^{\prime \prime }+3 y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 52

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

\[ y = c_{1} x^{{1}/{4}} \left (1-\frac {3}{5} x +\frac {1}{10} x^{2}-\frac {1}{130} x^{3}+\frac {3}{8840} x^{4}-\frac {3}{309400} x^{5}+\frac {3}{15470000} x^{6}-\frac {9}{3140410000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3}{14} x^{2}-\frac {3}{154} x^{3}+\frac {3}{3080} x^{4}-\frac {9}{292600} x^{5}+\frac {9}{13459600} x^{6}-\frac {1}{94217200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {9 x^7}{3140410000}+\frac {3 x^6}{15470000}-\frac {3 x^5}{309400}+\frac {3 x^4}{8840}-\frac {x^3}{130}+\frac {x^2}{10}-\frac {3 x}{5}+1\right )+c_2 \left (-\frac {x^7}{94217200}+\frac {9 x^6}{13459600}-\frac {9 x^5}{292600}+\frac {3 x^4}{3080}-\frac {3 x^3}{154}+\frac {3 x^2}{14}-x+1\right ) \]