Solve Heat Equation Urgently: No F(x)G(t) Assumption

[PeterPoon]

Heat equation! Urgent

Is anyone who are able to solve heat equation:
$${\frac{{\delta u}}{\delta t} = c^2\frac{{\delta^2 u}}{{\delta x^2}}$$

where,u(x,t)

I have looked for so many textbooks but most of them using the same method to solve which is "Assume u(x,t) = F(x)G(t) and sub this into the eqt. to solve.

Actually i am doing a Maths project right now and i have to solve this heat equation but without assuming u(x,t) = F(x)G(t), i have no idea to do it... i knew that it can be solved by simple differential equation technique, so i am here asking your help. Please help me. It is quite urgent! There is a attached file which is done by me and i want the similar solution without assuming u(x,t) =F(x)G(t).

 

[kurt.physics]

The solution to the heat equation without assuming u(x,t) = F(x)G(t) is to use the method of separation of variables. This involves expressing the solution as a product of two functions, one of which depends only on x and the other on t. Then, the equation can be separated into two equations: one for each function. The solution is then the product of the solutions to the two equations. For example, if we let u(x,t) = X(x)T(t), then we can separate the equation as follows:{\frac{{dX}}{dx} = -\frac{c^2}{T(t)} \frac{{d^2T}}{{dt^2}} {\frac{{dT}}{dt} = \frac{c^2}{X(x)} \frac{{d^2X}}{{dx^2}} Solving these two equations for X(x) and T(t) and then substituting them back into the original equation gives the general solution.