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Question
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
Solution
We have,
\[y = e^x..........(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = e^x\]
⇒ \[\frac{dy}{dx} = y............\left[\text{Using (1)}\right]\]
It is the given differential equation.
Here, \[y = e^x\] 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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