Solve the 3rd order partial differential equation by using the direct computation method.
Understand the Problem
The question asks for a solution to a third order partial differential equation using the direct computation method. This involves applying specific mathematical techniques to find the solution of the given equation.
Answer
The general solution is $$ u(x) = -\cos(x) + \frac{C_1}{2} x^2 + C_2 x + C_3 $$.
Answer for screen readers
The general solution to the third order partial differential equation is: $$ u(x) = -\cos(x) + \frac{C_1}{2} x^2 + C_2 x + C_3 $$
Steps to Solve
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Write the equation clearly We start with the equation for the third order partial differential equation: $$ u'''(x) = \sin(x) - \int_0^{\frac{\pi}{2}} x t u''(t) , dt $$
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Integrate the equation To solve for $u''(x)$, we integrate both sides with respect to $x$: $$ u''(x) = \int \left( \sin(x) - \int_0^{\frac{\pi}{2}} x t u''(t) , dt \right) dx $$
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Evaluate the integral Evaluate the integral of $\sin(x)$ and set up the expression: $$ u''(x) = -\cos(x) + C_1 - \int_0^{\frac{\pi}{2}} x t u''(t) , dt $$ where $C_1$ is a constant of integration.
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Integrate again Next, integrate once more to find $u'(x)$: $$ u'(x) = \int u''(x) , dx $$ This will give us: $$ u'(x) = -\sin(x) + C_1 x + C_2 + \int \int_0^{\frac{\pi}{2}} x t u''(t) , dt , dx $$ where $C_2$ is another constant of integration.
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Final integration Integrate for the third time to find $u(x)$: $$ u(x) = \int u'(x) , dx $$ This results in: $$ u(x) = -\cos(x) + \frac{C_1}{2} x^2 + C_2 x + C_3 $$
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Consider boundary conditions Finally, use any given boundary conditions to solve for the constants $C_1$, $C_2$, and $C_3$.
The general solution to the third order partial differential equation is: $$ u(x) = -\cos(x) + \frac{C_1}{2} x^2 + C_2 x + C_3 $$
More Information
The solution incorporates constants of integration that can be determined if specific boundary conditions are provided. The integration steps involve the sine function and require careful evaluation of the integral involving $u''(t)$.
Tips
- Forgetting to properly apply integration constants during integration stages.
- Neglecting to evaluate boundary conditions to find specific values for the constants.
- Miscalculating the integrals, especially with composite integrals present in the problem.
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