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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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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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 ...
Education
- MSc, Osmania University
- MSc, Indian Institute of Technology - Roorkee (I.I.T.-ROORKEE)
- BSc, Banaras Hindu University
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