Suppose that y′=ky, y(0)=7, and y′(0)=4. What is y as a function of t ?
PLEASE EXPLAIN
To solve the given differential equation, we can use the method of separation of variables. The differential equation is given as:
y' = ky
To solve for y as a function of t, we need to rearrange the equation and integrate both sides. The equation can be rewritten as:
y' / y = k
Now, integrate both sides with respect to t:
∫(1/y) dy = ∫k dt
Integrating the left side gives:
ln|y| = kt + C
Where C is the constant of integration.
Taking the exponential of both sides:
e^(ln|y|) = e^(kt + C)
The exponential of ln|y| simplifies to |y|:
|y| = e^(kt) * e^C
Since e^C is a constant, we can write it as A:
|y| = A * e^(kt)
To get rid of the absolute value, we can split the equation into two cases: y > 0 and y < 0.
For y > 0, the equation becomes:
y = Ae^(kt)
For y < 0, let's introduce another constant B as -A to avoid any confusion:
y = -Be^(kt)
Now, we need to substitute the given initial conditions to determine the values of A and B.
First, substitute t = 0 into the equation:
y(0) = A e^(k * 0) = A
Since y(0) = 7, we have A = 7 for y > 0.
Next, substitute the second initial condition, y'(0) = 4:
y' = kAe^(kt)
y'(0) = kAe^(k * 0) = kA
Since y'(0) = 4, kA = 4, and since A = 7, we have:
k(7) = 4
Solving for k, we find k = 4/7.
Finally, our solution is:
For y > 0: y = 7e^(4/7 t)
For y < 0: y = -7e^(4/7 t)
Therefore, y as a function of t is given by:
y(t) =
7e^(4/7 t) if t >= 0
-7e^(4/7 t) if t < 0
To find y as a function of t, we can solve the given differential equation using separation of variables. Here's a step-by-step explanation of how to do it:
Step 1: Rewrite the differential equation in the form dy/dt = ky.
Step 2: Separate the variables by bringing all terms involving y to one side and all terms involving t to the other side. It should look like:
1/y dy = k dt.
Step 3: Integrate both sides with respect to their respective variables. The left side integrates to ln|y| (natural logarithm of the absolute value of y), and the right side integrates to k * t + C, where C is the constant of integration.
Step 4: Solve for y by exponentiating both sides of the equation. This simplifies ln|y| to just |y|. The equation becomes:
|y| = e^(k * t + C).
Step 5: Remove the absolute value by considering two cases: y > 0 and y < 0.
Case 1: If y > 0, the equation becomes:
y = e^(k * t + C).
Case 2: If y < 0, the equation becomes:
-y = e^(k * t + C), which can be rewritten as:
y = -e^(k * t + C).
Step 6: Use the initial condition y(0) = 7 to find the value of the constant C. Substitute t = 0 and y = 7 into the equation from step 5. This will give you an equation to solve for C.
7 = e^(0 * k + C), which simplifies to:
7 = e^C.
Take the natural logarithm of both sides to solve for C:
ln(7) = C.
Step 7: Substitute the value of C back into the equations from step 5 to get the final solution for y as a function of t.
Case 1 (for y > 0): y = e^(k * t + ln(7)).
Case 2 (for y < 0): y = -e^(k * t + ln(7)).
Note that the value of k is missing in the given information. Once you have the value of k, you can substitute it into the equations to get the complete solution for y as a function of t.
dy/dt = ky
dy/y = k dt
ln y = kt + ln c
y = c*e^(kt)
Now plug in your values to determine c and k.