How do you solve the differential equation given f'(x)=4x, f(0)=6?

Ariel Wilkinson
2022-09-29
Answered

How do you solve the differential equation given f'(x)=4x, f(0)=6?

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Deanna Randolph

Answered 2022-09-30
Author has **3** answers

Substitute $\frac{dy}{dx}$ for f'(x)

$\frac{dy}{dx}=4x$

Use the separation of variable technique:

$dy=4xdx$

Integrate both sides:

$y=2{x}^{2}+C$

Evaluate C, using the initial condition:

$6=2{\left(0\right)}^{2}+C$

$C=6$

$\frac{dy}{dx}=4x$

Use the separation of variable technique:

$dy=4xdx$

Integrate both sides:

$y=2{x}^{2}+C$

Evaluate C, using the initial condition:

$6=2{\left(0\right)}^{2}+C$

$C=6$

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$\frac{dm}{dt}=a{m}^{0.75}[1-{\left(\frac{m}{M}\right)}^{0.25}]$

where $m$ is a function in terms of $t$ and $a$, $M$ are constants.

The end result (the integrated expression):

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where ${m}_{0}$ is another constant (in the context of the problem, the initial mass of organism when $t=0$).

I have tried solving by the integrating factors method but I couldn't express it in the form given... Anyone has any ideas?

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