Classical Equations and Boundary Value Problems

 $U\left(x,t\right)=\left[a\cos\left(\lambda x\right)+b\sin\left(\lambda x\right)\right]e^{-\lambda^2t}$  is a general solution for

• Wave Equation

• Heat Equation

• Laplace Equation

In wave equation, what is the notation for the initial velocity of a string?

•  $U_x\left(x,0\right)$  

•  $U_x\left(0,t\right)$  

•  $U_t\left(x,0\right)$  

•  $U_t\left(0,t\right)$  

 $U\left(x,t\right)=\left[b_2\cos\left(\lambda x\right)+b_1\sin\left(\lambda x\right)\right]\left[c_1\cos\left(\alpha\lambda t\right)+c_2\sin\left(\alpha\lambda t\right)\right]$  is a general solution for

• Heat  Equation

• Laplace Equation

• Wave Equation

 $U_{tt}=\alpha^2U_{xx}$  is a classical equation for ____________ equation.

Given that a rod is insulated at both ends and its lateral side.

Based on the above statement, determine the type of classical equation and its case.

• Heat Equation, Case A

• Laplace Equation, Case B

• Laplace Equation, Case A

• Heat Equation, Case B

Given $\int_{-1}^1\ \sin\left(2\pi x\right)\ \sin\left(\frac{n\pi x}{3}\right)\ dx\ =\ 0\ when\ n\ne m\ \ or\ \ 3\ when\ n=m$ . 

Determine the value of m.

m=____________

Given that a plate is kept at zero degree temperature at both vertical ends.

Based on the above statement, determine the type of classical equation and its case.

• Laplace Equation, Case A

• Laplace Equation, Case B

• Heat Equation, Case A

• Heat Equation, Case B

The general equation $U\left(x,t\right)=\left[a_1\cos\left(\lambda x\right)+a_2\sin\left(\lambda x\right)\right]\left[a_3\cos\left(3\lambda t\right)+a_4\sin\left(3\lambda t\right)\right]$ is the solution for which classical equation?

•   $3U_{xx}=U_t$  

•   $U_{tt}=9U_{xx}$ 

•  $9U_{xx}=U_t$  

•  $U_{tt}=3U_{xx}$  

 $U_{xx}+U_{yy}=0\$  is a classical equation of  ___________ equation.

Which is the classical equation used to find the displacement of a string?

• $U_t=kU_{xx}$

• $U_{xx}=-U_{yy}$

• $\frac{1}{\alpha^2}U_{tt}=U_{xx}$