# If \dfrac{dx}{dt}=5t. Determine x as a function of the time if x = 2 at t = 0.

## Question:

If {eq}\dfrac{dx}{dt}=5t {/eq}.

Determine x as a function of the time if x = 2 at t = 0.

## Initial Value Problem:

In the initial value problem, a first order differential equation is given having first order derivative. To solve this differential equation, we have to integrate this equation single time. Then with the help of given conditions, we find out the value of integral constant.

Reverse power rule is used to integrate polynomial functions having power in real numbers. We call it reverse rule because it is the reverse process of power rule of differentiation.

{eq}\int x^n = \frac{x^{n+1}}{n+1} + C {/eq}

where, C is the integral constant.

Given:

{eq}\dfrac{dx}{dt}=5t \\ \Rightarrow {dx}=5t \ dt {/eq}

Integrating this equation both sides:

{eq}\Rightarrow \int {dx}=5 \int t \ dt \\...

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