Let $y=e^{\pi x}$ be the solution of given differential equation,
$\Rightarrow \quad \frac{d y}{d x}=m e^{n x} \Rightarrow \frac{d^{2} y}{d x^{2}}=m^{2} e^{m x}$
$\therefore \quad 25 \frac{d^{2} y}{d x^{2}}-10 \frac{d y}{d x}+y=0$
$\Rightarrow \quad 25 \mathrm{m}^{2} e^{m x}-10 m e^{m x}+e^{m x}=0$
$\Rightarrow \quad e^{\operatorname{mux}}\left(25 m^{2}-10 m+1\right)=0$
$\Rightarrow$ Auxiliary equation
$\Rightarrow \quad(5 m-1)^{2}=0$
$\Rightarrow \quad m=\frac{1}{5}, \frac{1}{5}$
since, roots are real and euqal.
$\therefore$ General solution is $y=\left(c_{1}+c_{2} x\right) e^{x / 5} \ldots .$
$$
\begin{array}{l}
y(0)=1 \Rightarrow c_{1}=1 \\
y(1)=2 e^{1 / 5} \Rightarrow 2 c^{1 / 5}=\left(c_{1}+c_{2}\right) e^{1 / 5}
\end{array}
$$
$\Rightarrow \quad c_{1}+c_{2}=2 \Rightarrow c_{1}=1$
Putting the value of $c_{1}$ and $c_{2}$ in Eq. (i), we get particular solution
$$
y=(1+x) e^{x / 5}
$$