show that all solutions of 2y' + ty = 2 approach a limit as t--> ∞ and find the limiting value.
Can you please include all the steps and be clear so i can see what is going on?
Thank you so much for your time and help.
Can you please include all the steps and be clear so i can see what is going on?
Thank you so much for your time and help.
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There are two ways to solve the problem. One is integrating factor, and the other is variation of parameter. Personally, variation of parameter is easy to use because it is basically a formula. So I am going to show you how to use the formula.
y' + p(t)y = g(t)
A'(t) = g(t)exp(int(p(t))), where exp is exponential int is integral.
Then you need to integrate both sides and get A(t). Don't forget the constant C.
y(t) = A(t)exp(-int(p(t))).
In your case, you need to divide every thing by 2, and then you will have p(t) = t/2, and g(t) = 1. After that, you can use the formula.
A'(t) = exp(int(t/2)) = exp((t^2)/4)
A(t) = int(exp((t^2)/4)) + C (*since exp((t^2)/4) is not integrable unless you use Taylor series to rewrite the expression in polynomial form and integrate, which I don't recommend you to do that.)
y(t) = (exp(-int(t/2)))(int(exp((t^2)/4)) + C) = (exp(-t^2/4))(int(exp((t^2)/4)) + C)
When t → ∞, exp(-t^2/4) = 0, therefore, y(t) is approaching to 0 when t approaches to infinity.
y' + p(t)y = g(t)
A'(t) = g(t)exp(int(p(t))), where exp is exponential int is integral.
Then you need to integrate both sides and get A(t). Don't forget the constant C.
y(t) = A(t)exp(-int(p(t))).
In your case, you need to divide every thing by 2, and then you will have p(t) = t/2, and g(t) = 1. After that, you can use the formula.
A'(t) = exp(int(t/2)) = exp((t^2)/4)
A(t) = int(exp((t^2)/4)) + C (*since exp((t^2)/4) is not integrable unless you use Taylor series to rewrite the expression in polynomial form and integrate, which I don't recommend you to do that.)
y(t) = (exp(-int(t/2)))(int(exp((t^2)/4)) + C) = (exp(-t^2/4))(int(exp((t^2)/4)) + C)
When t → ∞, exp(-t^2/4) = 0, therefore, y(t) is approaching to 0 when t approaches to infinity.