12.13.36 problem 36

Internal problem ID [1927]
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 : 36
Date solved : Monday, January 27, 2025 at 05:38:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 x y^{\prime }+\left (4 x^{2}+2\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 )&=3\\ y^{\prime }\left (0\right )&=6 \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)+(2+4*x^2)*y(x)=0,y(0) = 3, D(y)(0) = 6],y(x),type='series',x=0);
 

\[ y = 3+6 x -3 x^{2}-5 x^{3}+x^{4}+\frac {31}{20} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 28

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

\[ y(x)\to \frac {31 x^5}{20}+x^4-5 x^3-3 x^2+6 x+3 \]