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Show that the equation is a solution to the differential equation.?
Show that y = (2/3)e^(x) + e^(-2x) is a solution for the differential equation y' + 2y = 2e^(x)
1 Answer
- hfshawLv 79 years agoFavorite Answer
When you are asked to show that some function is a solution to a differential equation, all you have to do is differentiate the solution function, then plug that result (and the solution itself, if necessary) into the differential equation and show that the equation is satisfied.
Here, we are supposed to show that:
y(x) = (2/3)*exp(x) + exp(-2x)
is a solution to:
dy/dx + 2y = 2exp(x)
Differentiating the proposed solution:
dy/dx = (2/3)*exp(x) - 2*exp(-2x)
Plugging this and the expression for y(x) into the differential equation yields:
(2/3)*exp(x) - 2*exp(-2x) + 2*((2/3)*exp(x) + exp(-2x)) ?=? 2*exp(x)
(2/3)*exp(x) - 2*exp(-2x) + (4/3)*exp(x) + 2*exp(-2x)) ?=? 2*exp(x)
(6/3)*exp(x) ?=? 2*exp(x)
2*exp(x) = 2*exp(x)
so the differential equation is satisfied, and this is therefore a solution.
