IGNOU MTE 8 2026 - 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

________ _________ __________ ________ ______ _____ __________ _______ _____ _____ _____ _____.
___ _____ _________ ________ ____ _________ _____ _________.
______ ___ _____ _____ _____ ______ ___ ___ _____ __________ _____.
___ ____ ____ ________ ________ _______ _________ _____ __________ __________ __________ ________.
___ ______ _____ ___ ____.
__________ _________ ___ _____ ________ ____ ______ ___ _____ _______ ______.
_________ ____ ___ __________ _______ _________ _____ ____ _____ ___.
_______ ________ ____ _______ _______ __________ ______ __________ _________ ____.
_____ _________ _____ ________ ________ _____ __________.
___ _____ ________ __________ _____ ___ ___ _____ ___ _____ __________.
_________ ___ ______ ___ ___ ______ _________ ____ _________ ___ ___.
__________ ____ ___ ____ _______.
_________ ______ __________ _______ _________ _________ ____.
_____ ___ _________ _________ _____ _____ ___ _____ ___.
________ _____ ________ _____ ___.
___ ___ _________ _____ ________ ____ __________ ______ _________ _________.
_____ ____ ____ ________ ____ _________ __________ ______.
__________ ____ ______ ___.

IGNOU ACADEMY

Your IGNOU journey, made easier.