12.13.22 problem 25

Internal problem ID [1913]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 25
Date solved : Monday, January 27, 2025 at 05:38:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -2 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

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

\[ y = -2+3 \left (x +2\right )-\frac {9}{2} \left (x +2\right )^{2}+\frac {11}{6} \left (x +2\right )^{3}+\frac {5}{24} \left (x +2\right )^{4}+\frac {7}{20} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 46

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

\[ y(x)\to \frac {7}{20} (x+2)^5+\frac {5}{24} (x+2)^4+\frac {11}{6} (x+2)^3-\frac {9}{2} (x+2)^2+3 (x+2)-2 \]