### 8.3.4 Important Solutions for Substitution and Separation of Variables Approaches

After having solved the one-dimensional heat equation, we will briefly summarize some solutions to similar problems, which, as we will see, we can apply fairly often. In the following we assume the function sought to be f (x). We will also discuss the various cases of boundary conditions that may apply.

#### 8.3.4.1 Characteristic Polynomials of Type a = λ: General Solution

For all values of λ the general solution for this problem is

$f\left(x\right)={c}_{0}{\mathrm{e}}^{\lambda x}$

(Eq. 8.83)

irrespective of the boundary conditions. See section 8.2.3 for details on this.

#### 8.3.4.2 Characteristic Polynomials ...