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$T(t)=e^{-\alpha^2n^2\pi^2t/l^2}$

$XT^{\prime}=\alpha^2 X^{\prime\prime}T$

$\dfrac{T^{\prime}}{\alpha^2 T}=-\dfrac{n^2\pi^2}{l^2}$

$\dfrac{X^{\prime\prime}}{X}=-\dfrac{n^2\pi^2}{l^2}$

$\mbox{None of the above}$

$\mbox{All of the above}$

$X(x) = \sin \left (\dfrac{n \pi x}{l} \right )$

	$H_1 \ \mbox{is a vector space and} \ H_2 \ \mbox{is a vector space.}$
	$H_1 \ \mbox{is a vector space, but} \ H_2 \ \mbox{is not a vector space.}$
	$H_1 \ \mbox{is not a vector space, but} \ H_2 \ \mbox{is a vector space.}$
	$H_1 \ \mbox{is not a vector space and} \ H_2 \ \mbox{is not a vector space.}$

	$y_1(t) \ \mbox{is a constant multiple of} \ y_2(t) \ \mbox{on} \ I.$
	$y_2(t) \ \mbox{is a constant multiple of} \ y_1(t) \ \mbox{on} \ I.$
	$\hspace{-6mm} y_1(t) \ \mbox{is not a constant multiple of} \ y_2(t) \ \mbox{on} \ I \ \mbox{and}\\y_2(t) \ \mbox{is not a constant multiple of} \ y_1(t) \ \mbox{on} \ I.$
	$\hspace{-6mm} y_1(t) \ \mbox{is a constant multiple of} \ y_2(t) \ \mbox{on} \ I \ \mbox{and}\\y_2(t) \ \mbox{is a constant multiple of} \ y_1(t) \ \mbox{on} \ I.$
	$y_2(t) \ \mbox{is not a constant multiple of} \ y_1(t) \ \mbox{on} \ I.$
	$y_1(t) \ \mbox{is not a constant multiple of} \ y_2(t) \ \mbox{on} \ I.$

	$\mbox{unstable if at least one eigenvalue of} \ A \ \mbox{has positive real part.}$
	$\mbox{unstable if all the eigenvalues of} \ A \ \mbox{have positive real part.}$
	$\mbox{unstable if at least one eigenvalue of} \ A \ \mbox{has negative real part.}$
	$\mbox{None of the above}$
	$\mbox{unstable if all the eigenvalues of} \ A \ \mbox{have negative real part.}$

Test!

	$\mbox{All of the above}$
	$\begin{cases} f_1(t, \xi_1, \xi_2)=0\\[2mm] f_2(t, \xi_1, \xi_2)=0 \end{cases}$
	$\mbox{None of the above}$
	$x_1(t)\equiv \xi_1, \ x_2(t)\equiv \xi_2 \ \mbox{is a solution of the system above}$

	$y=c_1e^{-t/2}\sin\left (\dfrac{\sqrt{3}t}{2}\right )+ c_2e^{-t/2}\cos\left (\dfrac{\sqrt{3}t}{2}\right )+t^2-\dfrac{1}{2}t$
	$y=c_1e^{-t/2}\sin\left (\dfrac{\sqrt{3}t}{2}\right )+ c_2e^{-t/2}\cos\left (\dfrac{\sqrt{3}t}{2}\right )+4t^2-3t$
	$y=c_1e^{-t/2}\sin\left (\dfrac{\sqrt{3}t}{2}\right )+ c_2e^{-t/2}\cos\left (\dfrac{\sqrt{3}t}{2}\right )+t^2-t$
	$\mbox{None of the above}$
	$y=c_1e^{-t/2}\sin\left (\dfrac{\sqrt{3}t}{2}\right )+ c_2e^{-t/2}\cos\left (\dfrac{\sqrt{3}t}{2}\right )+2t^2-3t$

	$y=c_1e^{-2x}+c_2e^x-\dfrac{1}{10}e^x\sin x-\dfrac{3}{10}e^x\cos x$
	$\mbox{None of the above}$
	$y=c_1e^{-2x}+c_2e^x-\dfrac{1}{4}e^x\sin x-\dfrac{3}{4}e^x\cos x$
	$y=c_1e^{-2x}+c_2e^x-\dfrac{2}{3}e^x\sin x-\dfrac{1}{3}e^x\cos x$
	$y=c_1e^{-2x}+c_2e^x-\dfrac{3}{5}e^x\sin x-\dfrac{7}{5}e^x\cos x$

	$\mbox{None of the above}$
	$\begin{cases} \dfrac{dx_1}{dt}&=x_1\\[2mm] \dfrac{dx_2}{dt}&=x_2\\[2mm] \dfrac{dx_3}{dt} &=\cos t-\sin x_2-x_1 \end{cases} \hspace{20mm} x_1(0)=1, \ x_2(0)=0, \ x_3(0)=0$
	$\begin{cases} \dfrac{dx_1}{dt}&=x_1\\[2mm] \dfrac{dx_2}{dt}&=x_2\\[2mm] \dfrac{dx_3}{dt} &=\cos t-\sin x_1-x_2 \end{cases} \hspace{20mm} x_1(0)=1, \ x_2(0)=0, \ x_3(0)=0$
	$\begin{cases} \dfrac{dx_1}{dt}&=x_2\\[2mm] \dfrac{dx_2}{dt}&=x_3\\[2mm] \dfrac{dx_3}{dt} &=\cos t-\sin x_1-x_2 \end{cases} \hspace{20mm} x_1(0)=1, \ x_2(0)=0, \ x_3(0)=0$
	$\begin{cases} \dfrac{dx_1}{dt}&=x_2\\[2mm] \dfrac{dx_2}{dt}&=x_3\\[2mm] \dfrac{dx_3}{dt} &=\cos t-\sin x_2-x_1 \end{cases} \hspace{20mm} x_1(0)=1, \ x_2(0)=0, \ x_3(0)=0$

	${\bf p}_1, \ {\bf p}_2, \ {\bf p}_3 \ \mbox{are linearly dependent}$
	${\bf p}_2, \ {\bf p}_3, \ {\bf p}_4, \ {\bf p}_5 \ \mbox{are linearly dependent}$
	${\bf p}_3, \ {\bf p}_4, \ {\bf p}_5 \ \mbox{are linearly dependent}$
	$\mbox{All of the above}$
	${\bf p}_1, \ {\bf p}_2, \ {\bf p}_3, \ {\bf p}_4 \ \mbox{are linearly dependent}$
	$\mbox{None of the above}$

	$\left [ \begin{array}{c} b_1\\[2mm] b_2\\[2mm] 2b_1+b_2 \end{array} \right ] \ \mbox{where} \ b_1, b_2 \ \mbox{are all real numbers}$
	$\left [ \begin{array}{c} b_1\\[2mm] b_2\\[2mm] b_1+2b_2 \end{array} \right ] \ \mbox{where} \ b_1, b_2 \ \mbox{are all real numbers}$
	$\left [ \begin{array}{c} b_1\\[2mm] b_2\\[2mm] 3b_1+2b_2 \end{array} \right ] \ \mbox{where} \ b_1, b_2 \ \mbox{are all real numbers}$
	$\mbox{None of the above}$
	$\left [ \begin{array}{c} b_1\\[2mm] b_2\\[2mm] 2b_1+3b_2 \end{array} \right ] \ \mbox{where} \ b_1, b_2 \ \mbox{are all real numbers}$

	${\bf x}(t)=c_1\left [ \begin{array}{c} \cos2t-\sin 2t\\[2mm] \cos 2t \end{array} \right ]+c_2\left [ \begin{array}{c} \cos 2t+\sin 2t\\[2mm] \sin 2t \end{array} \right ]$
	${\bf x}(t)=c_1\left [ \begin{array}{c} \cos2t-\sin 2t\\[2mm] \cos 2t \end{array} \right ]+c_2\left [ \begin{array}{c} 2\cos 2t+\sin 2t\\[2mm] \sin 2t \end{array} \right ]$
	$\mbox{None of the above}$
	${\bf x}(t)=c_1\left [ \begin{array}{c} \cos2t-2\sin 2t\\[2mm] \cos 2t \end{array} \right ]+c_2\left [ \begin{array}{c} \cos 2t+\sin 2t\\[2mm] \sin 2t \end{array} \right ]$
	${\bf x}(t)=c_1\left [ \begin{array}{c} \cos2t-2\sin 2t\\[2mm] \cos 2t \end{array} \right ]+c_2\left [ \begin{array}{c} 2\cos 2t+\sin 2t\\[2mm] \sin 2t \end{array} \right ]$

	$1-\dfrac{2}{\pi^2}\left (\cos(\pi x)+\dfrac{1}{9}\cos(3\pi x) +\dfrac{1}{25}\cos(5\pi x)+\dfrac{1}{49}\cos(7\pi x)\right )+\ldots$
	$\dfrac{1}{2}-\dfrac{4}{\pi^2}\left (\cos(\pi x)+\dfrac{1}{9}\cos(3\pi x) +\dfrac{1}{25}\cos(5\pi x)+\dfrac{1}{49}\cos(7\pi x)\right )+\ldots$
	$\dfrac{1}{2}-\dfrac{4}{\pi^2}\left (\sin(\pi x)+\dfrac{1}{9}\sin(3\pi x) +\dfrac{1}{25}\sin(5\pi x)+\dfrac{1}{49}\sin(7\pi x)\right )+\ldots$
	$\mbox{None of the above}$
	$1-\dfrac{2}{\pi^2}\left (\sin(\pi x)+\dfrac{1}{9}\sin(3\pi x) +\dfrac{1}{25}\sin(5\pi x)+\dfrac{1}{49}\sin(7\pi x)\right )+\ldots$