# Differential Equations and Indefinite Integrals

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))

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#### Solution Preview

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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

Type your answer here:

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.