Find the solution of the given initial value problem.
ty'+(t+1)y=t y(ln2)=0 t < 0
I NEED HELP ASAP
Kiara (assuming you are not also Alex), if you have a question, it is much better to put it in as a separate post in <Post a New Question> rather than attaching it to a previous question, where it is more likely to be overlooked.
State your question in as exact terms as possible.
To find the solution of the given initial value problem, we will use the method of integrating factors.
First, let's rewrite the given initial value problem in the standard form:
dy/dt + (t+1)/t * y = 1
The integrating factor for this differential equation is defined as the exponential of the integral of the coefficient of y. In this case, the coefficient of y is (t+1)/t, so the integrating factor (denoted by μ) is given by:
μ = exp(∫(t+1)/t dt)
Simplifying the integral:
μ = exp(∫(1 + 1/t) dt)
= exp(t + ln|t| + C1)
= e^C1 * e^t * |t|
Now, we multiply both sides of the original equation by the integrating factor μ:
μ * dy/dt + μ * (t+1)/t * y = μ * 1
e^C1 * e^t * |t| * dy/dt + e^C1 * e^t * |t| * (t+1)/t * y = e^C1 * e^t * |t|
Simplifying further:
e^C1 * e^t * |t| * dy/dt + (e^C1 * e^t * |t| * (t+1)/t) * y = e^C1 * e^t * |t|
The left-hand side of the equation can now be recognized as the derivative of the product of y(t) and e^C1 * e^t * |t|, using the product rule:
d/dt (y * (e^C1 * e^t * |t|)) = e^C1 * e^t * |t|
Now, we integrate both sides of the equation with respect to t:
∫ d/dt (y * (e^C1 * e^t * |t|)) dt = ∫ e^C1 * e^t * |t| dt
This simplifies to:
y * (e^C1 * e^t * |t|) = ∫ e^C1 * e^t * |t| dt
Integrating the right-hand side:
y * (e^C1 * e^t * |t|) = e^C1 * e^t * ∫ |t| dt
The integral of |t| can be split into two cases:
Case 1: t > 0
In this case, |t| simplifies to t, and the integral becomes:
y * (e^C1 * e^t * |t|) = e^C1 * e^t * ∫ t dt
= e^C1 * e^t * (t^2/2) + C2
Case 2: t < 0
In this case, |t| simplifies to -t, and the integral becomes:
y * (e^C1 * e^t * |t|) = e^C1 * e^t * ∫ -t dt
= -e^C1 * e^t * (t^2/2) + C2
Now, we can combine both cases into a single expression:
y * (e^C1 * e^t * |t|) = e^C1 * e^t * (t^2/2) + C2 * sgn(t)
Dividing both sides of the equation by e^C1 * e^t * |t|:
y = (t^2/2) + C2 * sgn(t) / (e^C1 * e^t * |t|)
Finally, we can substitute the initial condition y(ln2) = 0 into the obtained solution to solve for the constant C2:
0 = (ln2^2/2) + C2 * sgn(ln2) / (e^C1 * e^(ln2) * |ln2|)
0 = (ln2^2/2) + C2 * sgn(ln2) / (e^C1 * 2 * ln2)
0 = (ln2^2/2) + C2 * sgn(ln2) / (2 * e^C1 * ln2)
C2 * sgn(ln2) = -ln2^2 / (2 * e^C1 * ln2)
C2 = -ln2 / (2 * e^C1)
Finally, substituting this value of C2 back into the solution, we get the final solution of the given initial value problem:
y = (t^2/2) - (ln2 / (2 * e^C1)) * sgn(t) / (e^C1 * e^t * |t|)