# `f'(x) - 6x, f(0) = 8` Find the particular solution that satisfies the differential equation.

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### 2 Answers

To find the solution of the differential equation `f'(x)=6x`

` ` with initial condition `f(0)=8 ` , integrate both sides:

`int(f'(x) dx)=int(6x dx) =>f(x)+c_1=3x^2+c_2` ,where `c_1,c_2 ` are constants. This means that `f(x)=3x^2+c_2-c_1=>f(x)=3x^2+c ` . Using the initial condition `f(0)=3(0)^2+c=8=>c=8=>f(x)=3x^2+8 ` . Below we have plotted the differential equation and the solution.

First, find the integral

`int(6x)=f(x)=3x^2+C `

Solve for the constant using the given point `f(0)=8`

`3(0)^2+C=8`

`C=8`

Thus,

`f(x)=3x^2+8`