This is the initial value problem.
$5y''+ 2y'+ 7y = 0, \qquad y(0)=2, \quad y'(0)=1.$
Solution:
$ y = e^{-\frac{x}{5}} \cdot \left[2\cos\left(\frac{\sqrt{34}}{5} x\right)+ \frac{7}{\sqrt{34}}\sin\left(\frac{\sqrt{34}}{5}x\right)\right] $
Question: Which is the smallest T, for $|y(x)|\leq 0.1$ for all x > T.
I think I should isolate the x term in the the equation:
$ 0.1 = e^{-\frac{x}{5}} \cdot \left[2\cos\left(\frac{\sqrt{34}}{5} x\right)+ \frac{7}{\sqrt{34}}\sin\left(\frac{\sqrt{34}}{5}x\right)\right] $
But I don't know how to isolate x. Someone knows how to accomplish this?
Answer: T = 14.5115
$$ |y| = e^{-\frac{x}{5}} \cdot \left |\bigg[2\cos\left(\frac{\sqrt{34}}{5} x\right)+ \frac{7}{\sqrt{34}}\sin\left(\frac{\sqrt{34}}{5}x\right)\right]\bigg|$$
$$ = e^{-\frac{x}{5}} \sqrt {4+49/34}$$
Now you want $$ e^{-\frac{x}{5}} \sqrt {4+49/34}<0.1$$
Can you solve it from here?