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Solve:
$$x^{\prime \prime}=y, x(0)=3, x^{\prime}(0)=1$$
$$y^{\prime \prime}=x, y(0)=1, y^{\prime}(0)=-1$$

$$x^{\prime \prime}=y \rightarrow x^{\prime \prime}-y=0 \rightarrow D^{2} x-y=0 \cdots (1)$$

$$y^{\prime \prime}-x=0 \rightarrow-x+y^{\prime \prime}=0 \rightarrow-x+D^{2} y=0 \cdots (2)$$ multiply $$(2)$$ with $$D^{2}$$

$$-D^{2} x+D^{4} y=0 \quad \cdots (2)$$

$$D^{2} x-y=0$$
$$-D^{2} x+D^{4} y=0$$

$$\rightarrow D^{4} y-y=0 \rightarrow\left(D^{4}-1\right) y=0$$

$$\left(m^{4}-1\right) y=0 \quad y \neq 0 \quad m^{4}-1=0$$

$$\left(m^{2}+1\right)\left(m^{2}-1\right)=0 \longrightarrow\left(r^{2}-1\right)\left(r^{2}+1\right)=0$$

$$r_{1}=-1, r_{2}=+1, r_{3}=\pm i$$

$$y(t)=c_{1} e^{t}+c_{2} e^{-t}+c_{3} \cos (t)+c_{4} \sin (t)$$
$$y^{\prime}(t)=c_{1} e^{t}-c_{2} e^{-t}-c_3 \sin (t)+c_{4} \cos (t)$$
$$y^{\prime \prime}(t)=c_{1} e^{t}+c_{2}e^{-t}-c_{3} \cos (t)-c_{4} \sin (t)$$

$$y(t)=c_{1} e^{t}+c_{2}e^{-t}+c_{3} \cos (t)+c_4 \sin (t)$$

$$x(t)=c_{1} e^{t}+c_{2}e^{-t}-c_{3} \cos (t)-c_4 \sin (t)$$

$$3=c_{1}+c_{2}-c_{3}$$
$$1=c_{1}+c_{2}+c_3$$

$$x^{\prime}(t)=c_{1} e^{t}-c_{2} e^{-t}+c_{3} \sin (t)-c_4 \cos (t)$$

$$y^\prime(t)=c_{1} e^{t}-c_2 e^{-t}-c_3 \sin (t)+c_4 \cos (t)$$

at $$\quad t=0 \rightarrow$$

$$1=c_{1}-c_{2}-c_{4}$$
$$-1=c_{1}-c_{2}-c_{3}$$

Solving the equations:

$$\begin{array}{ll}{c_{1}=1} & {c_{3}=-1} \\ {c_{2}=1} & {c_{4}=-1}\end{array}$$

$$x(t)=e^{t}+e^{-t}+\cos (t)+\sin (t)$$

$$y(t)=e^{t}+e^{-t}-\cos (t)-\sin (t)$$

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