x and time t of a particular heat equation is: U(x,t) = [Cn(e-6tn^2)sin(nx) where the sum goes from 1 to ∞. A) Write out the heat equation that has this solution including the boundary conditions. B) If U(x,0) = 3sin(2x) -5sin(4x) find the two non-zero coefficients Cn C) Find U(x,t).

I get how to solve it as an exact equation, but I'm not seeing how to get to the supplied expression

please fill all the spaces

The equation in differential form M dx + Ñ dy = 0 is not exact. Indeed, we have M₁ - Ñ , For this exercise we can find an integrating factor which is a function of x alone since M₁ - Ñ , I Ñ = N = can be considered as a function of x alone. Namely we have μ(x) Multiplying the original equation by the integrating factor we obtain a new equation M dx + N dy = 0 where M Which is exact since My N₂ (3y + 2xe-³¹) dx + (1 − 2ye¯³¹)dy = 0 = = = = are equal. This problem is exact. Therefore an implicit general solution can be written in the form F(x, y) = C where F(x, y) = Finally find the value of the constant C so that the initial condition y(0) = 1. C =