Answer
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Hint:
We see that the given differential equation is the second order differential equation. It means that we differentiate the equation \[y = A\cos nx + B\sin nx\] two times to obtain the second order differential equations. The differentiation is a method of finding a function that generates the rate of change between one variable and another variable. In this equation \[y = A\cos nx + B\sin nx\], y is the dependent variable and x is the independent variable.
Complete step by step solution:
The equation given in the problem is as follows.
\[y = A\cos nx + B\sin nx\].
We can differentiate the above equation with respect to x by using Chain rule.
$\dfrac{d}{{dx}}\left( y \right) = \dfrac{d}{{dx}}\left( {A\cos nx + B\sin nx} \right)\\
\dfrac{{dy}}{{dx}} = - A\sin nx\left( {\dfrac{d}{{dx}}\left( {nx} \right)} \right) + B\cos nx\left( {\dfrac{d}{{dx}}nx} \right)\\
= - nA\sin nx + Bn\sin nx$
The required differential equation needs the second order of differential equation therefore, we will again differentiate the above equation with respect to x.
$\dfrac{{{d^2}y}}{{d{x^2}}} = - An\cos nx\left( {\dfrac{d}{{dx}}\left( {nx} \right)} \right) + Bn\sin nx\left( {\dfrac{d}{{dx}}nx} \right)\\
= - {n^2}A\cos nx - B{n^2}\sin nx\\
= - {n^2}\left( {A\cos nx + B\sin nx} \right)$
We know that \[y = A\cos nx + B\sin nx\] which can be used in the above equation. So, substitute the value of \[\left( {A\cos nx + B\sin nx} \right)\] with y in the above expression.
$\dfrac{{{d^2}y}}{{d{x^2}}} = - {n^2}\left( y \right)\\
\Rightarrow \dfrac{{{d^2}y}}{{d{x^2}}} + {n^2}y = 0$
Hence, the above result proves the required equation is \[\dfrac{{{d^2}y}}{{d{x^2}}} + {n^2}y = 0\]. Thus, it is proved that \[y = A\cos nx + B\sin nx\] is a solution of the differential equation: $\dfrac{{{d^2}y}}{{d{x^2}}} + {n^2}y = 0$.
Additional Information:
A differential equation is like an equation of dependent terms differentiated to the different orders. There are a lot of ways of solving such types of equations.
Note:
Make sure to use proper chain rules while doing the differentiation. You should know the basic formula of the trigonometry such as the differential of \[\sin x\] is \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\] and the differential of \[\cos x\] is \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]. Also don’t get confused with the process of differentiation and the process of the integration.
We see that the given differential equation is the second order differential equation. It means that we differentiate the equation \[y = A\cos nx + B\sin nx\] two times to obtain the second order differential equations. The differentiation is a method of finding a function that generates the rate of change between one variable and another variable. In this equation \[y = A\cos nx + B\sin nx\], y is the dependent variable and x is the independent variable.
Complete step by step solution:
The equation given in the problem is as follows.
\[y = A\cos nx + B\sin nx\].
We can differentiate the above equation with respect to x by using Chain rule.
$\dfrac{d}{{dx}}\left( y \right) = \dfrac{d}{{dx}}\left( {A\cos nx + B\sin nx} \right)\\
\dfrac{{dy}}{{dx}} = - A\sin nx\left( {\dfrac{d}{{dx}}\left( {nx} \right)} \right) + B\cos nx\left( {\dfrac{d}{{dx}}nx} \right)\\
= - nA\sin nx + Bn\sin nx$
The required differential equation needs the second order of differential equation therefore, we will again differentiate the above equation with respect to x.
$\dfrac{{{d^2}y}}{{d{x^2}}} = - An\cos nx\left( {\dfrac{d}{{dx}}\left( {nx} \right)} \right) + Bn\sin nx\left( {\dfrac{d}{{dx}}nx} \right)\\
= - {n^2}A\cos nx - B{n^2}\sin nx\\
= - {n^2}\left( {A\cos nx + B\sin nx} \right)$
We know that \[y = A\cos nx + B\sin nx\] which can be used in the above equation. So, substitute the value of \[\left( {A\cos nx + B\sin nx} \right)\] with y in the above expression.
$\dfrac{{{d^2}y}}{{d{x^2}}} = - {n^2}\left( y \right)\\
\Rightarrow \dfrac{{{d^2}y}}{{d{x^2}}} + {n^2}y = 0$
Hence, the above result proves the required equation is \[\dfrac{{{d^2}y}}{{d{x^2}}} + {n^2}y = 0\]. Thus, it is proved that \[y = A\cos nx + B\sin nx\] is a solution of the differential equation: $\dfrac{{{d^2}y}}{{d{x^2}}} + {n^2}y = 0$.
Additional Information:
A differential equation is like an equation of dependent terms differentiated to the different orders. There are a lot of ways of solving such types of equations.
Note:
Make sure to use proper chain rules while doing the differentiation. You should know the basic formula of the trigonometry such as the differential of \[\sin x\] is \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\] and the differential of \[\cos x\] is \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]. Also don’t get confused with the process of differentiation and the process of the integration.
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