Question

If \[y = a{x^{n + 1}} + b{x^{ - n}}\], then \[{x^2}\frac{{dy}}{{d{x^2}}}\] equals to
A. \[n(n - 1)y\]
B. \[n(n + 1)y\]
C. \[ny\]
D. \[{{\rm{n}}^2}{\rm{y}}\]

Answer 1

Hint :
The second-order derivative is the derivative of a function's first derivative. We have to differentiate the assumed solution and find the arbitrary constants. Substitute the derivatives in the given differential equation. 
Formula use:
\[y = {x^n}\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = n{x^{n-1}}\]
Complete step-by-step solution:
In the question, we have been given the equation
\[y = a{x^{n + 1}} + b{x^{ - n}}\]--- (1)
We have to differentiate the equation (1) with respect to \[x\], then we obtain
\[\frac{{{\rm{d}}y}}{{\;{\rm{d}}x}} = a(n + 1){x^n} - bn{x^{ - (n + 1)}}\]---- (2)
Now, we have to differentiate the equation (2), in order to get the second order differential equation:
\[\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{{{\rm{d}}y}}{{\;{\rm{d}}x}}} \right) = \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {a(n + 1){x^n} - bn{x^{ - (n + 1)}}} \right)\]
This can be rewritten as,
\[\frac{{{{\rm{d}}^2}y}}{{\;{\rm{d}}{x^2}}} = a(n + 1)n{x^{n - 1}} - bn[ - (n + 1)]{x^{ - n - 1 - 1}}\]
From the above equation, rewrite it by taking \[n\] outside to make it less complicated:
\[ \Rightarrow \frac{{{{\rm{d}}^2}y}}{{\;{\rm{d}}{{\rm{x}}^2}}} = an(n + 1){x^{n - 1}} + bn(n + 1){x^{ - n - 2}}\]
From the above equation, rewrite it by taking \[(n + 1)\] outside
\[ \Rightarrow \frac{{{{\rm{d}}^2}y}}{{\;{\rm{d}}{x^2}}} = n(n + 1)\left[ {a{x^{n - 1}} + b{x^{ - n - 2}}} \right]\]
Now, multiply both sides by \[{x^2}\].
\[{x^2}\frac{{{{\rm{d}}^2}y}}{{\;{\rm{d}}{x^2}}} = {x^2}n(n + 1)\left[ {a{x^{n - 1}} + b{x^{ - n - 2}}} \right]\]
\[ = n(n + 1)\left[ {a{x^2}{x^{n - 1}} + b{x^2}{x^{ - n - 2}}} \right]\]
Now, the obtained equation becomes as below:
Since, \[\left[ {{a^m}{a^n} = {a^{m + n}}} \right]\]
\[ = n(n + 1)\left[ {a{x^{2 + n - 1}} + b{x^{2 - n - 2}}} \right]\]
From the above equation, simplify the exponents:
\[ = n(n + 1)\left[ {a{x^{n + 1}} + b{x^{ - n}}} \right]\]
We have to rewrite the resultant equation as below.
Since we know that, \[\left[ {y = a{x^{n + 1}} + b{x^{ - n}}} \right]\]
\[ = n(n + 1)y\]
Therefore, the value of \[{x^2}\frac{{{{\rm{d}}^2}y}}{{\;{\rm{d}}{x^2}}}\] is \[n(n + 1)y\].
Hence, the option B is correct.
Note:
Students mostly make mistakes by improper usage of formulas. Finding the general solutions to second-order homogeneous differential equations is the first thing we need to learn. The kind of roots we uncover for the differential equation will determine the formula we use for the general solution. We will first substitute the function y in terms of the variable r in order to find the roots. Initial conditions will be given to us so that we can find the specific solution to the differential equation by accounting for constants.