I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial x}(1,t)+u(1,t)=0,&t>0,\\ u(x,0)=1,&0<x<1.\end{matrix}\right.$$ Starting as usual with $u(x,t)=X(x)T(t)$, I get that $X$ has to verify
$$\left\{\begin{matrix}X''+\lambda X=0,\\ X(0)=0,\\ X'(1)+X(1)=0.\end{matrix}\right.$$ then I show that $\lambda>0$ and after I get this equation $$\frac{\tan(\sqrt{\lambda})}{\sqrt{\lambda}}=-1$$ and here is where I am stuck.