54.9.10 problem 10

Internal problem ID [8680]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 10
Date solved : Monday, January 27, 2025 at 04:19:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+\left (1+4 x \right ) 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: 41

Order:=8; 
dsolve(2*x^2*diff(y(x),x$2)-x*(1+2*x)*diff(y(x),x)+(1+4*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} \sqrt {x}\, \left (1-3 x +\frac {1}{2} x^{2}+\frac {1}{30} x^{3}+\frac {1}{280} x^{4}+\frac {1}{2520} x^{5}+\frac {1}{23760} x^{6}+\frac {1}{240240} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {2}{3} x +\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 70

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

\[ y(x)\to c_2 \sqrt {x} \left (\frac {x^7}{240240}+\frac {x^6}{23760}+\frac {x^5}{2520}+\frac {x^4}{280}+\frac {x^3}{30}+\frac {x^2}{2}-3 x+1\right )+c_1 \left (1-\frac {2 x}{3}\right ) x \]