Question 1

A thin, insulated rod of length $L=\pi$ has its ends maintained at $0^{\circ}$ C. Initially, the temperature distribution along the rod is given by $u(x,0)=\sin(3x)$ . The temperature $u(x,t)$ satisfies the heat equation $u_t=\alpha u_{xx}$ , where $\alpha$ is the thermal diffusivity. Which of the following represents the temperature distribution $u(x,t)$ for $t>0$ ?

Accepted Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Question 2

A string of length $L=\pi$ is fixed at both ends. Its initial displacement is zero, and its initial velocity is given by $0^{\circ}$ . The displacement $u(x,0)=\sin(3x)$ satisfies the wave equation $u(x,t)$ . Find the displacement $u_t=\alpha u_{xx}$ for $\alpha$ .

Accepted Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Question 3

A botanist is studying the growth of a unique flower. The shape of its petals is modeled by a function $L=\pi$ that satisfies the differential equation: $0^{\circ}$ . The botanist wants to understand the petal's curvature near its base $u(x,0)=\sin(3x)$ . Assuming a series solution of the form $u(x,t)$ what is the recurrence relation for the coefficients $u_t=\alpha u_{xx}$ ?

Accepted Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

Answer

$e^{-9\alpha t}\sin(3x)$

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