Question

What is a solution to the differential equation $ \dfrac{{dy}}{{dx}} = y? $

Answer 1

Hint: As we know that the differential equation is an equation which contains one or more derivatives, where derivatives are terms describing the rate of change of quantities that vary in them continuously. In general we can say that the solution of differential equations is an equation that expresses the functional independence of one variable or more. We know that it typically includes a constant term that is not present in the original differential equation. We can represent it by $ C $ .

Complete step by step solution:
As per the above question we have $ \dfrac{{dy}}{{dx}} = y $ . Here we have the function $ 'y' $ and we know that the derivative of $ 'y' $ is equal to $ y $ itself.
We know that there is also one function of this derivative which is $ y = {e^x} $ . And the derivative of this function is also equal to itself $ {e^x} $ .
So we can say that $ y = {e^x} $ is one solution to this differential equation.
Now let us assume that $ dy $ and $ dx $ are two discrete variables. So it can be written as $ dy = ydx $ .
Now we divide both sides by $ y $ , so we have
 $
\dfrac{{dy}}{y} = \dfrac{{ydx}}{y} \\
 \Rightarrow \dfrac{{dy}}{y} = dx
 $ .
We will now integrate the left hand side and right hand side: $ \int {\dfrac{1}{y}dy = \int {dx} } $ .
On further solving we have $ \ln \left| y \right| = x + C $ .
Now we can raise both the sides of the equation by $ e $ to cancel the $ \ln $ ,
So it can be written as $ y = \pm {e^{x + C}} $ , by taking the constant in the front we have
 $ y = \pm C{e^x} $ .
Since we know that $ C $ can be either positive and negative, we can eliminate the positive and negative sign i.e. $ y = C{e^x} $ .
Hence is the differential equation of the given question.
So, the correct answer is “ $ y = C{e^x} $ ”.

Note: WE should note that we have to add the constant the integration at the end always i.e. $ C $ . Here in this question we need only one. Before solving this kind of question we should be fully aware of the integration and their methods . We know that any constant multiple of $ {e^x} $ is a solution to the differential equation.