Question

If y=y(x) satisfied the differential equation $8\sqrt{x}\left( \sqrt{9+\sqrt{x}} \right)dy={{\left( \sqrt{4+\sqrt{9+\sqrt{x}}} \right)}^{-1}}dx$, x>0 and $y\left( 0 \right)=\sqrt{7}$, then $y\left( 256 \right)$ is equal to?
(a) 16
(b) 80
(c) 3
(d) 9

Answer 1

Hint: First, we can clearly see that in this question variables x and y are easily separable. Then, by using the substitution as $\sqrt{4+\sqrt{9+\sqrt{x}}}=t$ to get the differentiation of the terms by using the chain rule, we get the value of integral. Then, to get the value of c, by substituting the $y\left( 0 \right)=\sqrt{7}$ which means at x=0,we get the value of c and then we get the required result.

Complete step-by-step answer:
In this question, we are supposed to find the value of $y\left( 256 \right)$when y=y(x) satisfied the differential equation $8\sqrt{x}\left( \sqrt{9+\sqrt{x}} \right)dy={{\left( \sqrt{4+\sqrt{9+\sqrt{x}}} \right)}^{-1}}dx$, x>0 and $y\left( 0 \right)=\sqrt{7}$.
So, before proceeding for this, we must rewrite the following question in normal form by changing the inverse power as:
$8\sqrt{x}\left( \sqrt{9+\sqrt{x}} \right)dy=\dfrac{1}{\left( \sqrt{4+\sqrt{9+\sqrt{x}}} \right)}dx$
So, we can clearly see that in this question variables x and y are easily separable as:
$dy=\dfrac{1}{8\sqrt{x}\left( \sqrt{9+\sqrt{x}} \right)\left( \sqrt{4+\sqrt{9+\sqrt{x}}} \right)}dx$
Now, by using the substitution as $\sqrt{4+\sqrt{9+\sqrt{x}}}=t$ to get the differentiation of the terms by using the chain rule as:
$\dfrac{1}{2\sqrt{4+\sqrt{9+\sqrt{x}}}}\times \dfrac{1}{2\sqrt{9+\sqrt{x}}}\times \dfrac{1}{2\sqrt{x}}dx=dt$
Then, by substituting the value calculated above in the given integral as:
$dy=dt$
Then, by integrating both sides where c is constant, we get:
$y=t+c$
Then, by again substituting the value of t as assumed in the above mentioned statement as:
$y=\sqrt{4+\sqrt{9+\sqrt{x}}}+c$
Now, to get the value of c, by substituting the $y\left( 0 \right)=\sqrt{7}$which means at x=0, we get $y=\sqrt{7}$:
$\begin{align}
  & \sqrt{7}=\sqrt{4+\sqrt{9+\sqrt{0}}}+c \\
 & \Rightarrow \sqrt{7}=\sqrt{4+\sqrt{9}}+c \\
 & \Rightarrow \sqrt{7}=\sqrt{4+3}+c \\
 & \Rightarrow \sqrt{7}=\sqrt{7}+c \\
 & \Rightarrow c=0 \\
\end{align}$
Then, by substituting the value of c as 0, so by getting the value of $y\left( 256 \right)$ as:
$\begin{align}
  & y=\sqrt{4+\sqrt{9+\sqrt{256}}} \\
 & \Rightarrow y=\sqrt{4+\sqrt{9+16}} \\
 & \Rightarrow y=\sqrt{4+\sqrt{25}} \\
 & \Rightarrow y=\sqrt{4+5} \\
 & \Rightarrow y=\sqrt{9} \\
 & \Rightarrow y=3 \\
\end{align}$
So, we get the value of $y\left( 256 \right)$as 3.
Hence, option (c) is correct.

Note: Now, to solve these types of questions we need to know some of the basic rules of differentiation which is chain rule. So, the concept of chain rule says that we will continue the differentiation till we reach the last value of x and then we get the value as required.