12.11.9 problem 19

Internal problem ID [1848]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 19
Date solved : Monday, January 27, 2025 at 05:37:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 40

Order:=6; 
dsolve([(1+x)*diff(y(x),x$2)+2*(x-1)^2*diff(y(x),x)+3*y(x)=0,y(1) = a__0, D(y)(1) = a__1],y(x),type='series',x=1);
 

\[ y = a_{0} +a_{1} \left (x -1\right )-\frac {3}{4} a_{0} \left (x -1\right )^{2}+\left (\frac {a_{0}}{8}-\frac {a_{1}}{4}\right ) \left (x -1\right )^{3}+\left (\frac {a_{0}}{16}-\frac {a_{1}}{48}\right ) \left (x -1\right )^{4}+\left (\frac {3 a_{0}}{64}+\frac {a_{1}}{40}\right ) \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 95

AsymptoticDSolveValue[{(1+x)*D[y[x],{x,2}]+2*(x-1)^2*D[y[x],x]+3*y[x]==0,{y[1]==a0,Derivative[1][y][1]==a1}},y[x],{x,1,"6"-1}]
 

\[ y(x)\to \frac {1}{20} (x-1)^5 \left (\frac {1}{4} \left (\frac {3 \text {a1}}{2}-\frac {3 \text {a0}}{4}\right )+\frac {9 \text {a0}}{8}+\frac {\text {a1}}{8}\right )+\frac {1}{12} (x-1)^4 \left (\frac {3 \text {a0}}{4}-\frac {\text {a1}}{4}\right )+\frac {1}{6} (x-1)^3 \left (\frac {3 \text {a0}}{4}-\frac {3 \text {a1}}{2}\right )-\frac {3}{4} \text {a0} (x-1)^2+\text {a0}+\text {a1} (x-1) \]