dydt=10−0.2y, y(0)=10.
a. Solve this differential equation,
I tried solving it and I keep on getting 10. I dont know how to approach this question
dy/dt = 10−0.2y
5 dy/dt = 50-y
dy/(50-y) = .2 dt
-ln(50-y) = .2t+c
y = ce^-.2t + 50
y(0) = 10, so
10 = c+50
c = -40
y = 50 - 40e^-.2t
Do I plug in 0 for t?
To solve this differential equation dy/dt = 10 - 0.2y, where y(0) = 10, we can use separation of variables method.
Step 1: Separate the variables by moving all the terms involving y to one side and all the terms involving t to the other side:
dy / (10 - 0.2y) = dt
Step 2: Integrate both sides with respect to their respective variables:
∫ (1 / (10 - 0.2y)) dy = ∫ dt
Step 3: Evaluate the integral:
Using the substitution u = 10 - 0.2y, we can rewrite the integral on the left-hand side as:
-5 ∫ (1/u) du
= -5 ln|u| + C
= -5 ln|10 - 0.2y| + C
And the integral on the right-hand side is simply t + C.
Step 4: Combine the results and solve for y:
-5 ln|10 - 0.2y| + C = t + C
-5 ln|10 - 0.2y| = t
ln|10 - 0.2y| = -t/5
Take the exponential of both sides:
10 - 0.2y = e^(-t/5)
Solve for y:
0.2y = 10 - e^(-t/5)
y = (10 - e^(-t/5)) / 0.2
Therefore, the solution to the differential equation is y = (10 - e^(-t/5)) / 0.2, where y(0) = 10.
To solve the given differential equation, you can use the method of separation of variables. Here's how you can approach it:
1. Given differential equation: dy/dt = 10 - 0.2y
2. Rearrange the equation to have all y terms on one side and all t terms on the other side:
dy/(10 - 0.2y) = dt
3. Now, you can integrate both sides of the equation. Integrate the left side with respect to y and the right side with respect to t:
∫(1/(10 - 0.2y))dy = ∫dt
4. Evaluate the integrals:
(1/-0.2)ln|10 - 0.2y| = t + C
5. Simplify the left side:
-5ln|10 - 0.2y| = t + C
6. Rewrite the equation in exponential form in order to isolate y:
ln|10 - 0.2y| = -(t + C)/5
7. Remove the natural logarithm by exponentiating both sides:
|10 - 0.2y| = e^(-(t + C)/5)
8. Eliminate the absolute value by considering the positive and negative cases:
10 - 0.2y = e^(-(t + C)/5) OR 10 - 0.2y = -e^(-(t + C)/5)
9. Solve for y in each case:
Case 1: 10 - 0.2y = e^(-(t + C)/5)
0.2y = 10 - e^(-(t + C)/5)
y = (10 - e^(-(t + C)/5))/0.2
Case 2: 10 - 0.2y = -e^(-(t + C)/5)
0.2y = 10 + e^(-(t + C)/5)
y = (10 + e^(-(t + C)/5))/0.2
10. Finally, to find the specific solution given the initial condition y(0) = 10, substitute t = 0 and y = 10 into the general solution obtained in step 9. Solve for the constant C by plugging in these values:
y(0) = (10 - e^(-0 - C)/5))/0.2 = 10
Now, you can solve for C using algebraic manipulation.