u_(t)=u_(xx)+t-2, 0lt xlt 1,tgt 0

This is a partial differential equation (PDE) problem. To solve it, we need to find the solution \( u(x,t) \) that satisfies the given PDE and any boundary or initial conditions.The given PDE is: To solve this, we can use the method of separation of variables. However, before proceeding with that method, let's first consider if there are any boundary conditions or initial conditions provided. Since none are given, we'll assume the problem is to find the general solution.First, we assume a solution of the form \( u(x,t) = X(x)T(t) \). Substituting this into the PDE gives: Dividing both sides by \( X'(x)T(t) \) gives: For the left-hand side to be a constant, the right-hand side must also be a constant. Let's denote this constant by : Now, we solve the spatial part \( X''(x) - \lambda X'(x) = 0 \): This is a second-order linear homogeneous differential equation. The general solution is: Next, we solve the temporal part \( T'(t) - (\lambda - \frac{t-2}{X'(x)T(t)}) = 0 \): This is a first-order nonlinear ordinary differential equation. To solve it, we can use an integrating factor or separate variables, but it's more straightforward to assume a solution of the form \( T(t) = e^{\int (\lambda - \frac{t-2}{X'(x)T(t)}) dt} \).However, without specific boundary or initial conditions, we can't proceed further with a unique solution. The general approach involves solving the spatial and temporal parts separately and then combining them to form the solution \( u(x,t) \).In summary, the general form of the solution to the given PDE is: where \( T(t) \) is a solution to the temporal part of the PDE.