15.22.10 problem 10

Internal problem ID [3344]
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 : 10
Date solved : Monday, January 27, 2025 at 07:34:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\sin \left (x \right ) \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: 0.002 (sec). Leaf size: 22

Order:=7; 
dsolve([diff(y(x),x$2)-y(x)=sin(x),y(0) = 1, D(y)(0) = 2],y(x),type='series',x=0);
 

\[ y \left (x \right ) = 1+2 x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{7}\right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 43

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

\[ y(x)\to \frac {x^6}{720}+\frac {x^5}{60}+\frac {x^4}{24}+\frac {x^3}{2}+\frac {x^2}{2}+2 x+1 \]