I came across this problem which says: Consider the equation: $\frac{dy}{dt}=(1+f^{2}(t))y(t); \quad t\geqslant 0,$ where $f$ is bounded continuous function on $[0,\infty)$.Then which of the following options is correct?
(a) This equation admits a unique solution $y(t)$ and further $\lim_{t\to\infty}y(t)$ exists and finite ,
(b) This equation admits 2 linearly independent solutions,
(c) This equation admits a bounded solution for which $\lim_{t\to\infty}y(t)$ does not exist,
(d) this equation admits a unique solution and further, $\lim_{t\to\infty}y(t)=\infty$.
I have taken $f(t)$ to be $1/(1+t)$ so that $f$ is bounded and continuous in the aforementioned interval and then applying the given conditions, i see that the option (d) holds true.
$ \int \frac{dy}{y}=\int (1+f^{2}(t))dt=\int (1+\frac{1}{(1+t)^{2}})dt=t-1/(1+t)+a. $ Hence, $y(t)=ce^{t}e^{-1/(1+t)}$, where $c=e^{a}$. Now we put the value of c and see that $y$ approaches to infinity as $t$ tends to infinity
Am i correct? Is there any other better way to approach the problem?Any kind of help will be highly appreciated.Thanks everyone in advance for your time.