12.13.38 problem 38

Internal problem ID [1929]
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 : 38
Date solved : Monday, January 27, 2025 at 05:38:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 y^{\prime \prime }+2 x y^{\prime }+\left (-x^{2}+4\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 )&=-2\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 20

Order:=6; 
dsolve([3*diff(y(x),x$2)+2*x*diff(y(x),x)+(4-x^2)*y(x)=0,y(0) = -2, D(y)(0) = 3],y(x),type='series',x=0);
 

\[ y = -2+3 x +\frac {4}{3} x^{2}-x^{3}-\frac {19}{54} x^{4}+\frac {13}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{3*D[y[x],{x,2}]+2*x*D[y[x],x]+(4-x^2)*y[x]==0,{y[0]==-2,Derivative[1][y][0] ==3}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {13 x^5}{60}-\frac {19 x^4}{54}-x^3+\frac {4 x^2}{3}+3 x-2 \]