Find a function f such that f(3) = 2 and (t^2 + 1)f^{\prime}(t) + (f(t))^2 + 1 = 0 for t not...

Question:

Find a function f such that f(3) = 2 and {eq}(t^2 + 1)f^{\prime}(t) + \left[f(t)\right]^2 + 1 = 0 {/eq} for {eq}t \neq 1 {/eq}.

Separable Differential equations:

This problem involves solving a given separable differential equation. A differential equation is said to be separable, we can isolate the variables to either side of the equation. For example, let's say we have an ODE as -

{eq}\displaystyle \frac{dy}{dx} = g(y) \times f(x) {/eq}

Then, we can separate the variables as -

{eq}\displaystyle \frac{dy}{g(y)} = f(x) dx {/eq}

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Given a function {eq}\displaystyle f(x) {/eq} such that,

{eq}\displaystyle f(3) = 2 {/eq}

And,

{eq}\displaystyle (t^2+1) f'(t) + ((f(t))^2 +...