12.13.34 problem 34

Internal problem ID [1925]
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 : 34
Date solved : Monday, January 27, 2025 at 05:38:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+5 x y^{\prime }-\left (-x^{2}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=6\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

Order:=6; 
dsolve([diff(y(x),x$2)+5*x*diff(y(x),x)-(3-x^2)*y(x)=0,y(0) = 6, D(y)(0) = -2],y(x),type='series',x=0);
 

\[ y = 6-2 x +9 x^{2}+\frac {2}{3} x^{3}-\frac {23}{4} x^{4}-\frac {3}{10} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 34

AsymptoticDSolveValue[{D[y[x],{x,2}]+5*x*D[y[x],x]-(3-x^2)*y[x]==0,{y[0]==6,Derivative[1][y][0] ==-2}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to -\frac {3 x^5}{10}-\frac {23 x^4}{4}+\frac {2 x^3}{3}+9 x^2-2 x+6 \]