7.13.24 problem 25

Internal problem ID [423]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.1 (Introduction). Problems at page 206
Problem number : 25
Date solved : Monday, January 27, 2025 at 02:53:33 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=y^{\prime }+y \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 18

Order:=6; 
dsolve([diff(y(x),x$2)=diff(y(x),x)+y(x),y(0) = 0, D(y)(0) = 1],y(x),type='series',x=0);
 

\[ y = x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{8} x^{4}+\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 33

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

\[ y(x)\to \frac {x^5}{24}+\frac {x^4}{8}+\frac {x^3}{3}+\frac {x^2}{2}+x \]