For the following initial value problem:
dy/dt=1/((y+1)(t-2)), y(0)=0
a)Find a formula for the solution.
b) State the domain of definition of the solution.
c) Describe what happens to the solution as it approaches the limit of its domain of definition. Why can't the solution be extended for more time?
I separated and integrated and got y(t)=sqrt(2ln|t-2|+C)-1 and I don't really know where to go from there.
Using the initial condition, C will be 2*ln|-2|
y(t)=sqrt(2 ln|t-2|+C)-1
If y is to be zero when t = 0 then
1 = sqrt (2 ln 2 +C)
C = 1 -2 ln 2 = -.386
y = sqrt (2 ln|t-2| -.386) - 1
You can not take ln of zero or a negative number so t may not be equal to 2
Try some solutions around t = t to see what happens to y
To find a formula for the solution, you correctly separated and integrated the differential equation:
dy/dt = 1/((y+1)(t-2))
To start, multiply both sides by (y + 1):
(y + 1)dy = dt/(t - 2)
Next, integrate both sides:
∫(y + 1)dy = ∫dt/(t - 2)
Integrating ∫(y + 1)dy gives:
(y^2/2 + y) = ln|t - 2| + C
Now, to find C, we can use the initial condition y(0) = 0:
(0^2/2 + 0) = ln|0 - 2| + C
0 = ln(2) + C
C = -ln(2)
Now, substitute C back into the equation:
(y^2/2 + y) = ln|t - 2| - ln(2)
Simplifying, we can combine the ln terms on the right side:
(y^2/2 + y) = ln(|t - 2|/2)
To solve for y, we can factor out a 2 from the left side:
2(y^2/2 + y) = ln(|t - 2|/2)
y^2 + 2y = ln(|t - 2|/2)
Now, isolate y:
y^2 + 2y - ln(|t - 2|/2) = 0
This is the explicit formula for the solution to the initial value problem.
Moving on to the domain of definition of the solution, we need to consider the values of t where the equation is defined. From the original differential equation, we can see that (t - 2) and (y + 1) cannot be equal to zero. This means that t ≠ 2 and y ≠ -1. Therefore, the domain of definition for the solution is all values of t except t = 2, and all values of y except y = -1.
Finally, we need to describe what happens to the solution as it approaches the limit of its domain of definition, and why it can't be extended for more time.
Looking at the explicit formula, we can see that the natural logarithm term ln(|t - 2|/2) will become undefined as t approaches 2. This causes a singularity in the equation, leading to the breakdown of the solution. As t approaches 2, the solution becomes unbounded, and the function y(t) is no longer defined. This is why the solution cannot be extended for more time.