Solve u_t equal k*u_x
To solve the equation u_t = k*u_x, we need to differentiate both sides of the equation.
Differentiating u_t with respect to t gives us:
u_tt = k*u_xt
And differentiating u_x with respect to x gives us:
u_tx = k*u_xx
Now we can substitute these derivatives back into the original equation:
u_tt = u_xt
Since the second mixed derivative is equal to the first mixed derivative, this equation simplifies to:
u_tt = u_xx
This is the wave equation, which has solutions of the form:
u(x,t) = f(x - ct) + g(x + ct)
where f and g are arbitrary functions and c is a constant.
To solve the equation u_t = k*u_x, where u is a function of two variables t and x, we can use the method of characteristics.
1. Define a new variable s = x - kt.
2. Differentiate both sides of the equation with respect to t and x:
du/dt = du/ds * ds/dt = du/ds * (-k)
du/dx = du/ds * ds/dx = du/ds
3. Substitute these expressions into the equation:
du/dt = k * du/dx
du/ds * (-k) = k * du/ds
4. Cancel out the common factor of du/ds:
-du/ds = du/ds
5. Rearrange the equation:
du/ds + du/ds = 0
6. Combine like terms:
2 * du/ds = 0
7. Divide both sides by 2:
du/ds = 0
8. Integrate both sides with respect to s:
∫ du/ds ds = ∫ 0 ds
u = C
9. Rewriting the variable in terms of t and x:
u(x, t) = C
where C is an arbitrary constant. Therefore, the general solution of the equation u_t = k*u_x is u(x, t) = C, where C is a constant.