Differential equation dy dx =y,y(0)=1Function y=e^x - Vidyadip

Question

Differential equation $\frac{\text{dy}}{\text{dx}}=\text{y},\text{y}(0)=1$Function $\text{y}=\text{e}^\text{x}$

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Answer

we have,$y=e^x$ ...(1) Differentiating both sides of (1)with respect to x, we get
$=\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}$
$=\frac{\text{dy}}{\text{dx}}=\text{y}$ [Using (1)]
It is the given differential equation.
Here, y = ex satisfies the given differential equation; hence, it is a solution.
Also, when $x = 0, y = e^0= 1$, i.e. y(0) = 1
Hence, $y = e^x$ is the solution to the given initial value problem.

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