10.14.3 problem 3

Internal problem ID [1385]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 3
Date solved : Monday, January 27, 2025 at 04:56:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y \ln \left (x \right )&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 16

Order:=6; 
dsolve([x^2*diff(y(x),x$2)+(1+x)*diff(y(x),x)+3*ln(x)*y(x)=0,y(1) = 2, D(y)(1) = 0],y(x),type='series',x=1);
 

\[ y = 2-\left (x -1\right )^{3}+\frac {7}{4} \left (x -1\right )^{4}-\frac {49}{20} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 30

AsymptoticDSolveValue[{x^2*D[y[x],{x,2}]+(1+x)*D[y[x],x]+3*Log[x]*y[x]==0,{y[1]==2,Derivative[1][y][1]==0}},y[x],{x,1,"6"-1}]
 

\[ y(x)\to -\frac {49}{20} (x-1)^5+\frac {7}{4} (x-1)^4-(x-1)^3+2 \]