12.13.33 problem 33

Internal problem ID [1924]
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 : 33
Date solved : Monday, January 27, 2025 at 05:38:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\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 )&=1\\ 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)-3*x*diff(y(x),x)+(5+2*x^2)*y(x)=0,y(0) = 1, D(y)(0) = -2],y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

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

\[ y(x)\to \frac {x^5}{3}-\frac {3 x^4}{8}+\frac {2 x^3}{3}-\frac {5 x^2}{2}-2 x+1 \]