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# Differential Equations and Indefinite Integrals

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1. Find the solution of the differential equation dy/dx=x^3 -x, with the initial condition y(0) = -3.

2. Find the solution of the differential equation dy/dt=t^2 / (3y^2), with the initial condition y(0) = 8.

3. Find the solution of the differential equation dy/dx=(x=2)y^(1/2), with the initial condition y(0) = 1.

4. Find the derivative of the function f(x) = ln(x + 1)/ln(2x + 1)

5. Evaluate the following indefinite integral: (e^x)(e^(e^x)) dx =

6. Evaluate the following expression: sec(arctan(10))

7. Find the derivative of the function: f(x) = arccot(tan(2x + 3))

https://brainmass.com/math/integrals/differential-equations-indefinite-integrals-303654

#### Solution Preview

Dear student, please refer to the attachment for the solutions.

Problem: Find the solution of the differential equation dy/dx=x^3 -x with the initial condition y(0) = -3.

Solution:
dy/dx=x^3 -x
Or,dy=(x^3 -x)dx
On Integrating,âˆ«â–’ã€–dy=âˆ«â–’(x^3 -x)dxã€—
Or,y=x^4/4-x^2/2+C
Given: y(0)= -3
Therefore,
-3=0^4/4-0^2/2+C
Or,-3=C
Therefore,
y=x^4/4-x^2/2-3
y(x) = x^4/4-x^2/2-3
Problem: Find the solution of the differential equationdy/dt=t^2 / (3y^2), with the initial condition y(0) = 8.
Solution:
dy/dt=t^2/((3y^2 ) )
Or,3y^2 dy=t^2 dt [cross-multiplication]
On integrating both sides,3âˆ«â–’ã€–y^2 dy=âˆ«â–’ã€–t^2 dtã€—ã€—
Or,3(y^3/3)=t^3/3+C
Or ...

#### Solution Summary

This solution is comprised of detailed step-by-step calculations and solutions of the given problems. The solution also provides students with a clear perspective of the underlying mathematical concepts behind solving Differential Equations and Indefinite Integrals in general.

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