Non-order equation

Dec 2015
1,084
169
Earth
\(\displaystyle y^{(1)} y^{(2)}\cdot .... \cdot y^{(n)} =e^{nx} \; \;\) , \(\displaystyle n\in \mathbb{N}\) .

Another way to write it better : \(\displaystyle \prod_{i=1}^{n} \frac{d^i y}{dx^i } =e^{nx}\) .
 
Dec 2015
1,084
169
Earth
Let \(\displaystyle b_{n} =e^{nx}\; \; \) then \(\displaystyle y^{(n+1)} =\frac{b_{n+1} }{b_n }=e^x\) .
The equation \(\displaystyle y^{(n+1)} =e^x \;\) has solution \(\displaystyle y=c+e^x \).