7.15.25 problem 25

Internal problem ID [481]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 25
Date solved : Monday, January 27, 2025 at 02:54:01 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{2} x +\frac {1}{8} x^{2}-\frac {1}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-x +\frac {1}{3} x^{2}-\frac {1}{15} x^{3}+\frac {1}{105} x^{4}-\frac {1}{945} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{3840}+\frac {x^4}{384}-\frac {x^3}{48}+\frac {x^2}{8}-\frac {x}{2}+1\right )+c_2 \left (-\frac {x^5}{945}+\frac {x^4}{105}-\frac {x^3}{15}+\frac {x^2}{3}-x+1\right ) \]