solve each of the following initial value problems and plot the solutions for serveral variables of ynaught
a)dy/dt = -y+5
I got to the point where I have:
y= ce^-t + 5
they give y(0)=y(naught) so I tried plugging in 0,0 and got c to be -5. but the answer they give in the back includes the y(naught) in the equation, so my answer is incorrect. why is this>
never mind I got it
To solve the given initial value problem, let's revisit the steps:
Given: dy/dt = -y + 5
1) Separate the variables: Move all terms involving y to one side and all terms involving t to the other side:
dy/(5 - y) = -dt
2) Integrate both sides:
∫dy / (5 - y) = -∫dt
3) Evaluate the integrals:
-ln|5 - y| = -t + C
4) Solve for y:
ln|5 - y| = t + C
5 - y = e^(t + C) (applying the natural logarithm properties)
y = 5 - e^(t + C)
6) Simplify:
y = 5 - Ce^t (where C = e^C)
Now, to find the value of the constant C, we can use the given initial condition y(0) = ynaught. Plugging in t = 0 and y = ynaught into the equation, we get:
ynaught = 5 - Ce^0
ynaught = 5 - C
Rearranging the equation, we have:
C = 5 - ynaught
Substituting C back into the equation, we get:
y = 5 - (5 - ynaught)e^t
This is the correct solution for the initial value problem dy/dt = -y + 5, which includes the ynaught term. The solution you obtained (y = ce^(-t) + 5) is missing the ynaught term, which explains why your answer differed from the one given in the back of the book.
To plot the solution for several values of ynaught, you can assign different numerical values to ynaught and plot the corresponding solution curves.