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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))

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

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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,∫▒〖dy=∫▒(x^3 -x)dx〗
    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,3∫▒〖y^2 dy=∫▒〖t^2 dt〗〗
    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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