Who can tell me what's the Deffrential equation of of this general equation y=(Ae^5x)+(Bxe^5x)
please 😊😊
Another way (if you do not know about the characteristic equation):
from $\displaystyle y= Ae^{5x}+ Bxe^{5x}$, $\displaystyle y'= 5Ae^{5x}+ Be^{5x}+ 5Bxe^{5x}= (5A+ B)e^{5x}+ 5Bxe^{5x}$ and $\displaystyle y''= 5(5A+ B)e^{5x}+ 5Be^{5x}+ 25Bxe^{5x}= (25A+ 10B)e^{5x}+ 25Be^{5x}$.
Subtracting 10 times the second equation from the third, $\displaystyle y''- 10y'= (25A+ 10B)e^{5x}+ 25Be^{5x}- (50A+ 10B)e^{5x}- 50Bxe^{5x}= -(25Ae^{5x}+ 25Bxe^{5x})$. Adding 25y eliminates the arbitrary constants, A and B, and gives $\displaystyle y''- 10y'+ 25y= 0$.