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Which of the following functions is a solution of a differential equation of the form

$$y'=f(y)$$

where $f$ is continuously differentable for all real $y$

(a) $y=sin(t)$

(b) $y=cos(t)$

(c) $y=t^2$

(d) $y=t^3-t$

(e) $y=cosh(t)$

(f) $y=tanh(t)$

What i tried

It is clear that the ODE must be an autonomous ODE. I picked $(c)$ because since

$$y=t^2$$ is a solution then $$y=t$$ and $$y=1$$ are solutions too.

Hence the solutions are $$y(t)=C_{1}+C_{2}(t)+C_{3}(t^2)$$

Differentiating the above expression gives

$$y'(t)=C_{2}+2_C{3}(t)$$

Which is of the form $$y'=f(y)$$ Im unsure of my answers though. Could anyone please explain. Thanks

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1 Answer

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Let's help with your example $y(t)=t^2$. Then $y'(t)=2t$ and expressing $t$ again in terms of $y$ gives $$y'(t)=2\sqrt{|y(t)|}.$$ Of course, this only works without problems for $t\ge0$.

If similar restrictions on the range of $t$ are allowed, then there exist such differential equations for all of the functions.

If no restriction on $t$ is permissible, then you need that the functional relation $y=y(t)$ is bijective and thus invertible, which would disqualify $y(t)=t^2$.

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