Can I have an analytical solution for $\frac{\partial \theta}{\partial t}=\frac{\partial^2\theta}{\partial {x}^2}+1$

Question

Subjected to the following boundary conditions:

$\left.\frac{\partial \theta}{\partial x}\right|_{x=0,t}=c_1\theta\bigg\vert_{x=0,t}$

$-\left.\frac{\partial \theta}{\partial x}\right|_{x=1,t}=c_2\theta\bigg\vert_{x=1,t}$

$\theta\bigg\vert_{x,t=0}=0$

Where $c_1$ and $c_2$ are nonzero constants.

I know I can solve it with fourier sin and cos expansion, but can I have analytical solution?

I accept "as far I know, it doesn't" or just tell me the method I should study.

(Physical meaning: It is a dimensionless transient heat conduction for 1D with uniform source and both boundary conditions with convection)

Answer 1

$\theta(x,t)=t$ is an analytic solution.

EDIT: With the introduction of the boundary conditions, I would say: No, I don't know any other method not involving Fourier transforms or separation of variables.