Answer: y(x) = (6/7)*e^(7*x) - (6/7)
Explanation:
I will use the notation:
dy/dx = y'
We have:
y' + 7*y = 6.
Because in the right side we do not have any term that depends on x, we can assume that y is an exponential function:
y = a*e^(b*x) + c
where a, b and c are constants.
y' = b*a*e^(b*x)
First, let's apply the condition in zero.
y(0) = 0 = a*e^(b*0) + c = a + c = 0.
then we have c = -a.
And we can write our equation as:
y = a*e^(b*x) - a.
Then the differential equation becomes:
b*a*e^(b*x) - 7*a*e^(b*x) + 7*a = 6.
In the left side we do not have any exponential, then we must have b = 7 (So the exponentials cancel eachother)
7*a*e^(7*x) - 7*a*e^(7*x) + 7*a = 6
7*a = 6
a = 6/7.
Our equation is:
y(x) = (6/7)*e^(7*x) - (6/7)