82.47.1 problem Ex. 1

Internal problem ID [18976]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 102
Problem number : Ex. 1
Date solved : Tuesday, January 28, 2025 at 12:41:15 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-x^{2}+x \right ) y^{\prime \prime }+4 y^{\prime }+2 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: 38

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

\[ y \left (x \right ) = c_{1} \left (1-\frac {1}{2} x +\frac {1}{10} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-60 x +120 x^{2}-120 x^{3}+60 x^{4}-12 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {x^2}{10}-\frac {x}{2}+1\right )+c_1 \left (\frac {1}{x^3}-\frac {5}{x^2}+5 x+\frac {10}{x}-10\right ) \]