Using the method of separation of variables, solve when

Question

Using the method of separation of variables, solve $u_{xt} = e^{-t} \cos x$ when

$$u(x, 0) = 0, \frac{\partial u}{\partial t}(0, t) = 0.$
b) Find the temperature in a bar of length $\ell$ with both ends insulated and with initial temperature in the rod being $\sin \frac{\pi x}{\ell}$ .

09 Jan 2026

Answer :

Word Count : 198

For the equation (u_{xt}=e^{-t}\cos x), integrate first with respect to (x): [ u_t=\int e^{-t}\cos x,dx=e^{-t}\sin x+f(t), ] where (f(t)) is an arbitrary function of (t). Integrating with respect to (t), [ u(x,t)=\int \big(e^{-t}\sin x+f(t)\big),dt=-e^{-t}\sin x+\int f(t),dt+g(x), ] where (g(x)) is an arbitrary function of (x). Using the initial condition _______ ________ __________ ____ ________ _____ _______ __________ ___.
___ __________ ______ _______ _______ ___ _____ ______.
______ ______ __________ ______ ____ ________ ___ ____ __________ __________.
______ _________ _____ _______ ___ _________ __________ ___ _____ ___.
______ ______ _________ ________ ____ __________ _____ ____ ________ ______ _______.
___ _____ ________ ___ ___ _____ ___.
___ _______ ___ ____ ____ _____ _______.
_______ ___ _________ ______ ___.
_____ ____ __________ _________ _______ _________ __________.
________ _______ _________ ________ ____ _____ ________ ______.
______ __________ ______ ____ __________ ____ __________ ______.
__________ _________ _____ _______ _____ __________ ______ __________ _______ ___.
______ ________ __________ __________ ______ _____ _________ ____ ___.
______ ______ __________ ______ ____ __________ ________ ______ ________ ______.
__________ __________ _________ __________ ______ ________ ______ __________.
_____ _____ ___ ________ ______ _________ ______ _______ _________ _________.
_________ ____ ________ ___ ____ _______ ____ ______ ___ _____ _____ ______.
Get Full Answer on WhatsApp