https://file.aiquickdraw.com/mathbot/file/1728327942456nvvpi4io.jpg

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.

Solve the partial differential equation: x^2 u_xx - t^2 u_tt = 0 for x > 0, t > 0, with boundary conditions u(x, 1) = f(x), u_t(x, 1) = g(x).

Learn how to solve the hyperbolic partial differential equation x^2 u_xx - t^2 u_tt = 0 using the method of separation of variables and Cauchy-Euler equations. Includes a step-by-step breakdown of boundary conditions and wave equation analysis. Suitable for advanced students in mathematics.

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