Question

Which of the following relation in x and y is general solution of the differential equation $\frac{dy}{dx}=y$

• A. $y=e^x$
• B. $y=e^x+c$
• C. $y=k{e}^x$
• D. $y=k{e}^x+c$

Answer 1

The correct option is C $y=k{e}^x$

We want to find the general solution of the differential equation $\frac{dy}{dx}=y$ There are methods to find the general solution of a differential equation. We will learn them in the coming topics. For this question, we just have to check which among the given options satisfy the conditions of a general solution of a differential equation. Those conditions are
1. It should satisfy the given differential equation
2. The relation should have as many arbitrary constants as the order of the differential equation. Order of $\frac{dy}{dx}=y$ is 1
We will go through the options and check if it satisfies both the conditions.
A) $y=e^x$ This relation has zero arbitrary constants. But the required number of arbitrary constants is 1. This can’t be a general solution for the given differential equation.
B) $y=e^x+c$
Number of arbitrary constant is one. This is equal to the required number of arbitrary constants. We will check if it satisfies the given DE $\frac{dy}{dx}=y$
$\frac{dy}{dx}=e^x\ne y$
$\Rightarrow$ Not a solution
C) $y=k{e}^x$
This has one arbitrary constant and it satisfies the equation $\frac{dy}{dx}=y$
$\Rightarrow$ General solution
D) This has two arbitrary constants which can’t be reduced to one
$\Rightarrow$ Not a general solution