My approach : the general solutions are of the form \(\displaystyle y=c_1 e^x +c_2 \; \) after substitution to the general equation we must find the pairs \(\displaystyle (c_1 , c_2 )\).

(eq1) \(\displaystyle c_1 e^{nx} +c_2 =c_{1}^{n} e^{nx} \; \;\) , \(\displaystyle (c_1 , c_2 )\)=?

Now it is not easy to find the pairs but it can be seen that for the pair \(\displaystyle (1,0)\) one of the solutions is \(\displaystyle y=e^x\) .

(eq1) has solutions only if \(\displaystyle c_2=0\) but a proof is needed.

After that the equation turns into \(\displaystyle c_1 =c_{1}^{n} \; \; \) , \(\displaystyle n>0\) .

How to continue from (eq1) ?