Question

The elimination of the arbitrary constants \[{\rm{A}},{\rm{B}}\] and \[{\rm{C}}\] from \[y = A + Bx + C{e^{ - x}}\] leads to the differential equation
A. \[{y^{\prime \prime \prime }} - {y^\prime } = 0\]
B. \[{y^{\prime \prime \prime }} - {y^{\prime \prime }} + {y^\prime } = 0\]
C. \[{y^{\prime \prime \prime }} + {y^{\prime \prime }} = 0\]
D. \[{y^{\prime \prime }} + {y^{\prime \prime }} - {y^\prime } = 0\]

Answer 1

Hint:
By differentiating the equation with respect to x and continuing to do so until we have the values of constants A, B and C, we may solve this problem. We will apply certain substitutions after we have the values, and this will produce the outcome. Equations with one or more function derivatives are known as differential equations. The dependent variable is the function or variable whose derivative appears in the equation and the independent variable is the derivative with regard to the variable.
Formula use:
\[y = {x^n}\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = n{x^{n-1}}\]
\[y = {e^{-x}}\]
\[\frac{{dy}}{{dx}} ={-e^{-x}}\]

Complete step-by-step solution:
In the question, we have been given the solution for the differential equation
\[y = A + Bx + C{e^{ - x}}\]
Where, A, B and C are arbitrary constants.
We have to differentiation the equation\[y\]with respect to \[x\], we obtain the first order differentiation:
\[ \Rightarrow {y^\prime } = b - c{e^{ - x}}\]
Again, differentiating with respect to \[x\], we obtain the second order differential equation:
 \[ \Rightarrow {y^{\prime \prime }} = c{e^{ - x}}\]
Now, again differentiate the obtained equation to get third order differential equation:
\[ \Rightarrow {y^{\prime \prime \prime }} = - c{e^{ - x}}\]
Now, substitute all the obtained values in the equation given to get the resultant equation:\[ \Rightarrow {y^{\prime \prime \prime }} + {y^{\prime \prime }} = - c{e^{ - x}} + c{e^{ - x}} = 0\]
Therefore, the elimination of the arbitrary constants \[{\rm{A}},{\rm{B}}\] and \[{\rm{C}}\] from \[y = A + Bx + C{e^{ - x}}\] leads to the differential equation \[{y^{\prime \prime \prime }} + {y^{\prime \prime }} = 0\]
Hence, the option C is correct.

Note:
Students should require complete focus in order to be answered correctly. An equation that connects one or more functions and their derivatives is called a differential equation. In general, it establishes a connection between the rates of physical quantities. Given that the second derivatives may be unclear; arbitrary constants are ones that can have any value and remain unchanged while the values of the other variables in the equation change.