52.2.12 problem 12

Internal problem ID [8263]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 12
Date solved : Monday, January 27, 2025 at 03:47:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (x +3\right ) y^{\prime }+7 x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 64

Order:=8; 
dsolve(x*diff(y(x),x$2)+(x+3)*diff(y(x),x)+7*x^2*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} \left (1-\frac {7}{15} x^{3}+\frac {7}{120} x^{4}-\frac {1}{150} x^{5}+\frac {11}{160} x^{6}-\frac {197}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (2 x^{2}-\frac {14}{15} x^{5}+\frac {7}{60} x^{6}-\frac {1}{75} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+4 x -3 x^{2}+4 x^{3}-4 x^{4}+\frac {547}{225} x^{5}-\frac {5329}{3600} x^{6}+\frac {7642}{7875} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.054 (sec). Leaf size: 96

AsymptoticDSolveValue[x*D[y[x],{x,2}]+(x+3)*D[y[x],x]+7*x^2*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {11 x^6}{160}-\frac {x^5}{150}+\frac {7 x^4}{120}-\frac {7 x^3}{15}+1\right )+c_1 \left (\frac {5539 x^6-10432 x^5+14400 x^4-14400 x^3+14400 x^2-14400 x+7200}{7200 x^2}-\frac {1}{120} \left (7 x^4-56 x^3+120\right ) \log (x)\right ) \]