Solve the 3rd order partial differential equation by using the direct computation method.

Steps to Solve

1. Identify the Equation The equation given appears to involve finding a solution for the 3rd order partial differential equation with a specific condition. The function is derived from $ u''(x) = \sin(x) $.

2. Construct the Integral Representation We use the integral representation of the solution based on the provided equation: $$ u(x) = \int_0^{\frac{\pi}{2}} x t u''(t) dt $$

3. Substitute the Given Function Derivative Now, substitute the expression for $ u''(t) = \sin(t) $ into the integral: $$ u(x) = \int_0^{\frac{\pi}{2}} x t \sin(t) dt $$

4. Evaluate the Integral Next, evaluate the integral:

• First, observe that the integral of $ t \sin(t) $ can be computed using integration by parts: Let $ v = t $, $ dv = dt $, and $ dw = \sin(t) dt $, then $ w = -\cos(t) $.

Thus, applying integration by parts: $$ \int t \sin(t) dt = -t \cos(t) + \int \cos(t) dt $$

5. Final Integration The integral simplifies to: $$ \int t \sin(t) dt = -t \cos(t) + \sin(t) + C $$

Evaluate this from 0 to $\frac{\pi}{2}$:

• At $t = 0$: $0 \cdot \cos(0) + \sin(0) = 0$
• At $t = \frac{\pi}{2}$: $- \frac{\pi}{2} \cdot \cos(\frac{\pi}{2}) + \sin(\frac{\pi}{2}) = 1$

Now combine and calculate: $$ u(x) = x \left( 1 - 0 \right) = x $$