Question

Discuss the nature of the solutions for \(00\) of the equation $$ \rho \frac{\partial^{2} u}{\partial t^{2}}+H \frac{\partial u}{\partial t}-K^{2} \frac{\partial^{2} u}{\partial x^{2}}=0 $$ with boundary conditions \(u(0, t)=u(\pi, t)=0\), if \(\rho, H\), and \(K\) are positive constants. 9\. Prove that if \(\lambda \geq 0\), the equations (10.94) and (10.95) have no solution other than the trivial one: \(A(x) \equiv 0\).

Short answer

Question: Discuss the nature of the solutions for the given partial differential equation and constraints. Partial Differential Equation: $$ \rho \frac{\partial^{2} u}{\partial t^{2}}+H \frac{\partial u}{\partial t}-K^{2} \frac{\partial^{2} u}{\partial x^{2}}=0 $$ Boundary Conditions: \(u(0, t)=u(\pi, t)=0\), Given Constraint: \(\lambda \geq 0\). Answer: The nature of the solutions for the given PDE is composed of functions with certain combinations of \(\rho\), \(H\), and \(K\). The boundary conditions and specified constraints limit the solutions. In the case of the inequality constraint, only the trivial solution is valid.

Step-by-step solution

Simplify the given PDE

First, let's simplify the given PDE by rewriting it in terms of operators. We have the following: $$ \rho \frac{\partial^2 u}{\partial t^2} + H \frac{\partial u}{\partial t} - K^2 \frac{\partial^2 u}{\partial x^2} = 0 $$ This can be rewritten as: $$ \rho u_{tt} + Hu_t - K^2 u_{xx} = 0 $$ where \(u_{tt}\), \(u_{t}\) and \(u_{xx}\) represent the second-order partial derivative with respect to \(t\), first-order partial derivative with respect to \(t\), and second-order partial derivative with respect to \(x\), respectively.

Apply the boundary conditions

Now, let's apply the boundary conditions to the simplified PDE. We have: $$ u(0, t) = u(\pi, t) = 0 $$ Since we are looking for solutions with \(0 < x < \pi\) and \(t > 0\), we need to find a function \(u(x, t)\) that satisfies the simplified PDE for these values while satisfying the boundary conditions.

Conclude the nature of solutions

Taking into consideration the given boundary conditions and the simplified PDE, we can conclude that the nature of the solutions are such that they only exist for a certain set of values of \(\rho\), \(H\), and \(K\). The general solution of the PDE consists of a combination of functions that depend on the time variable \(t\) and spatial variable \(x\). As for the given inequality stating \(\lambda \geq 0\), the equations (10.94) and (10.95) have no solution other than the trivial solution of \(A(x) \equiv 0\). This is because any non-trivial solution would require other values of \(\lambda\) that would not satisfy the inequality. In conclusion, the nature of the solutions for the given PDE is composed of functions with certain combinations of \(\rho\), \(H\), and \(K\). The boundary conditions and specified constraints limit our solutions, and in the case of the inequality constraint, only the trivial solution is valid.