Q. 10 - Calculus

Question

In the process of solving  $\frac{dy}{dx}=y$ by separation of variables, we obtain the equation $\left|y\right|=e^{x+C}$. After solving for $y$, this equation becomes $y=A{e}^x$. How is $A$ related to $C$? What happened to the absolute value?

Step-by-Step Solution

Answer

Ans:  $A$ and $C$ are related by the relation $A=e^C$.

since the exponential function is positive for all real values of the exponent. Hence, there is no need to put the modulus sign on the left-hand side. Thus there is no change in absolute value.

Step 1. Given information.

given equations,

     $\left|y\right|=e^{x+C}$  and  $\left|y\right|=A{e}^x$

Step 2. The differential equation d y d x = y is solved by applying the variable separable method. After separating the variables and integrating both sides yields

   $\left|y\right|=e^{x+C} =e^x.e^C$

In the above result, since $e^C$ is a constant, replace it with another constant $A$, so that the solution is written in compact form as $y=A{e}^x$. Thus, the two constants $A$ and $C$ are related by the relation $A=e^C$.

Step 3. About absolute value:

Next, observe that in the equation $\left|y\right|=A{e}^x$, the right-hand side is always positive, since the exponential function is positive for all real values of the exponent. Hence, there is no need to put the modulus sign on the left-hand side. So, the modulus sign has been dropped and the solution is written as $y=A{e}^x$, where $A$ is a positive constant.