2 years ago
Show that y = 2/3e^(x) + e^(-2x) is a solution of the differential equation y’ + 2y = 2e^(x)
1 Answers
Best Answer
bushticketscolorado Staff answered 2 years ago
bushticketscolorado Staff answered 2 years ago
Step 1
y=(2)/(3)e^(x)+e^(-2x)
Differentiate with respect to x
y'=(2)/(3)e^(x)-2e^(-2x)
Step 2
We have to verify the following differential equation:
y'+2y=2e^(x)
Substitute the expressions for y' and y, To get
[(2)/(3)e^(x)-2e^(-2x)]+2[(2)/(3)e^(x)+e^(-2x)]=2e^(x)
[(2)/(3)e^(x)-2e^(-2x)]+[(4)/(3)e^(x)+2e^(-2x)]=2e^(x)
(2)/(3)e^(x)-2e^(-2x)+(4)/(3)e^(x)+2e^(-2x)=2e^(x)
((2)/(3)+(4)/(3))*e^(x)=2e^(x)
Hence verified
Result:
Hint: y'=(2)/(3)e^(x)-2e^(-2x)