Here is problem I am trying to solve for my differential equations class. I have spent several hours trying to solve it and have not been successful. I have tried breaking up the fraction various ways and using different techniques of integration but can't seem to get a solution that is valid near $t=\frac{1}{2}$. Here is the problem;
Solve the IVP $$\frac{dy}{dt}= t + \frac{t}{t^2−1}y ; \quad y\left(\frac{1}{2}\right) = 6.$$
Hint: Be careful with your integrating factor as you need to describe a solution that is valid near $t = 1/2.$
I am trying to solve the problem in the form: $$\frac{dy}{dt} – \frac{t}{t^2−1}y = t $$ using an integrating factor as the hint suggests.
If anyone could suggest some further hints as to how to solve this problem I would appreciate it. I would like to figure it out without having the solution and steps given to me.
Best Answer
$$\frac{dy}{dt}= t + \frac{t}{t^2−1}y$$ In the present case the method of integrating factor isn't the simplest. Nevertheless we will use it as requested.
You wrote : I am trying to solve the problem in the form $\frac{dy}{dt}- \frac{t}{t^2−1}y=t$ using an integrating factor.
This is a bad start because the correct form to start is $$N(t,y)dt+M(t,y)dy=0$$ You should start with this form : $$\left(t + \frac{t}{t^2−1}y\right)dt-dy=0$$ Then one have to find an integrating factor $\mu(t,y)$ so that the equation $$\mu(t,y)\left(t + \frac{t}{t^2−1}y\right)dt-\mu(t,y)dy=0\quad\text{be exact}.$$
HINT:
Try the simplest cases of function $\mu(t,y)$. For example try $\mu$ function of $t$ only. You will find $$\mu=\frac{1}{\sqrt{t^2-1}}$$