Differential Equations 1 Question 5
5. If $y=y(x)$ satisfies the differential equation $8 \sqrt{x}(\sqrt{9+\sqrt{x}}) d y=\sqrt{4+\sqrt{9+\sqrt{x}}}^{-1} d x, \quad x>0 \quad$ and $y(0)=\sqrt{7}$, then $y(256)=$
(2017 Adv.)
(a) 16
(b) 3
(c) 9
(d) 80
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Answer:
Correct Answer: 5. (b)
Solution:
- $\frac{d y}{d x}=\frac{1}{8 \sqrt{x} \sqrt{9+\sqrt{x}} \sqrt{4+\sqrt{9+\sqrt{x}}}}$
$\Rightarrow \quad y=\sqrt{4+\sqrt{9+\sqrt{x}}}+c$
Now, $y(0)=\sqrt{7}+c$
$\Rightarrow \quad c=0$
$$ y(256)=\sqrt{4+\sqrt{9+16}}=\sqrt{4+5}=3 $$
