The given equation in the image is:

$x^2 u_{xx} - t^2 u_{tt} = 0, \quad x > 0, \quad t > 0,$ with the boundary conditions:

$u(x, 1) = f(x), \quad u_t(x, 1) = g(x).$

This is a hyperbolic partial differential equation (PDE), specifically a version of the wave equation with variable coefficients. The equation appears to describe wave-like behavior in both space $x$ and time $t$, but with terms that scale based on $x^2$ and $t^2$.

Step-by-step Solution:

To solve this PDE, we can employ the method of separation of variables or look for solutions in a more direct manner based on known results for wave equations with similar structure.

1. Look for a solution of the form $u(x,t) = X(x) T(t)$, where $X(x)$ is a function of $x$ and $T(t)$ is a function of $t$.

2. Substitute $u(x,t) = X(x) T(t)$ into the PDE: $x^2 (X''(x) T(t)) - t^2 (X(x) T''(t)) = 0.$

3. Simplify the equation: $X''(x) \cdot x^2 T(t) = X(x) T''(t) \cdot t^2.$

4. Separate variables: divide both sides by $X(x) T(t)$: $\frac{X''(x)}{X(x)} \cdot x^2 = \frac{T''(t)}{T(t)} \cdot t^2.$ Since the left-hand side is a function of $x$ only and the right-hand side is a function of $t$ only, both sides must equal a constant. Let the constant be $\lambda$. Thus, we have two ordinary differential equations (ODEs):

   • For $X(x)$: $X''(x) + \frac{\lambda}{x^2} X(x) = 0,$ which is a Cauchy-Euler equation.

   • For $T(t)$: $T''(t) + \frac{\lambda}{t^2} T(t) = 0,$ which is also a Cauchy-Euler equation.

5. Solve the ODE for $X(x)$: The general solution for the Cauchy-Euler equation is: $X(x) = C_1 x^{r_1} + C_2 x^{r_2},$ where $r_1, r_2$ are roots of the characteristic equation $r(r-1) + \lambda = 0$, i.e., $r = \frac{1 \pm \sqrt{1-4\lambda}}{2}$.

6. Solve the ODE for $T(t)$: Similarly, the solution for $T(t)$ is: $T(t) = D_1 t^{r_1} + D_2 t^{r_2}.$

7. Apply the boundary conditions $u(x, 1) = f(x)$ and $u_t(x, 1) = g(x)$ to determine the constants $C_1, C_2, D_1, D_2$.

Do you want a detailed continuation of this solution and how the constants are determined using boundary conditions?

Related Questions:

1. What is the physical interpretation of this type of PDE in terms of waves?
2. How can separation of variables be applied to non-homogeneous boundary conditions?
3. What is the significance of the Cauchy-Euler equation in solving variable coefficient PDEs?
4. How does the presence of the $x^2$ and $t^2$ factors affect the behavior of the solution compared to a standard wave equation?
5. What are the different methods to solve hyperbolic PDEs besides separation of variables?

Tip:

When solving PDEs, always check if a change of variables or symmetry can simplify the equation before proceeding with standard techniques like separation of variables.