The order of the differential equation whose general solution is given by \(\mathrm{y}=\mathrm{C}_{1} \mathrm{e}^{(\mathrm{x}+\mathrm{C}) 2}+\left(\mathrm{C}_{3}+\mathrm{C}_{4}\right) \cdot \sin \left(\mathrm{x}+\mathrm{C}_{5}\right)\), where \(\mathrm{C}_{1}\), \(\mathrm{C}_{2}, \mathrm{C}_{3}, \mathrm{C}_{4}, \mathrm{C}_{5}\) are arbitrary Constant is (A) 5 (B) 4 (C) 3 (D) 2

Understanding the general solution of a differential equation is key to grasping the fundamental concept of these mathematical expressions. A differential equation involves an unknown function and its derivatives, and its solution is an equation that satisfies the given differential equation for all values in its domain.

The given general solution in the exercise, \[y= C_1 e^{(x+C_2)^2}+(C_3+C_4) \cdot \sin(x+C_5)\], reveals various constants, labeled as \(C_1, C_2, C_3, C_4,\textrm{ and } C_5\). These constants illustrate the 'arbitrary' or 'undetermined' elements which, when specified, provide a particular solution to the differential equation. A general solution encompasses all possible particular solutions, defined by various combinations of the constants.

In general, the order of a differential equation corresponds to the highest derivative of the function present within the equation. The process to find the order is to differentiate the general solution repeatedly until we can't produce a new independent condition involving a new constant. The presence of multiple exponential and trigonometric functions within the solution indicates an underlying higher-order differential equation with corresponding derivatives, which substantiate its complexity and order.

Derivatives play a fundamental role in the realm of differential equations, as they represent the rate at which a function is changing at any given point. In our exercise, derivatives are utilized to determine the order of the underlying differential equation by differentiating the general solution successively.

The process involves taking the first derivative \[y'\], which gives us the change of \(y\) with respect to \(x\). Continuing to the second derivative \[y''\], and then the third \[y'''\], each step provides new terms that might include different constants. It's essential to recognize the pattern of how these constants appear with each differentiation and to identify at what stage no new constants appear.

Impact of Differentiating Exponential and Trigonometric Functions

Understanding how to differentiate functions like exponentials and trigonometric forms is crucial since the process results in new function types or altered coefficients. For instance, differentiating \(e^{(x+C_2)^2}\) introduces a polynomial factor, while differentiating \(\sin(x+C_5)\) switches between sine and cosine.

Solving differential equations is a systematic process that involves multiple steps to find the function that satisfies the equation. The solution often requires integration, which is the inverse process of differentiation, to return to the function from its derivative.

However, in this exercise, our goal was distinct. Instead of solving a differential equation for an unknown function, we verified the order of a differential equation given its general solution. This reverse-engineering method showcases an essential skill in understanding and manipulating differential equations. The key is to use differentiation to reveal how many independent arbitrary constants the solution contains because each constant often stems from an integration in solving the original differential equation.

The technique we applied requires differentiating the provided general solution and finding the point where differentiation does not introduce a new arbitrary constant. Here, we confirmed that it was unnecessary to solve the differential equation directly to establish its order, offering an excellent example of the theoretical aspect of differential equations.