15.22.11 problem 11

Internal problem ID [3345]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 40, page 186
Problem number : 11
Date solved : Monday, January 27, 2025 at 07:34:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y&={\mathrm e}^{2 x} \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 )&=0 \end{align*}

✓ Solution by Maple

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

Order:=7; 
dsolve([diff(y(x),x$2)-2*y(x)=exp(2*x),y(0) = 0, D(y)(0) = 0],y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{10} x^{5}+\frac {7}{180} x^{6}+\operatorname {O}\left (x^{7}\right ) \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 39

AsymptoticDSolveValue[{D[y[x],{x,2}]-2*y[x]==Exp[2*x],{y[0]==0,Derivative[1][y][0] ==0}},y[x],{x,0,"7"-1}]
 

\[ y(x)\to \frac {7 x^6}{180}+\frac {x^5}{10}+\frac {x^4}{4}+\frac {x^3}{3}+\frac {x^2}{2} \]