Today we will investigate how we may determine the nature of the original functi

Question

Today we will investigate how we may determine the nature of the original function just from knowledge of the rate of change of the function. Recall: An equation of the form dy/dx = some expression is a differential equation if, upon appropriate substitution of the function and/or its derivative(s), the differential equation is true. a) Verity that y = x^3 + C is a solution for the differential equation dy/dx = 3x^2. b) verify further that y = x^3 + 4 is a solution the differential equation dy/dx = 3x^2 with y(0) = 4. Definitions: The statement y(0) = 4 is an initial condition, and the differential equation and the initial condition are together an initial-value problem.

Explanation / Answer

From the given question,

part 1

a) dy/dx= 3x2

dy= 3x2 dx

integrating both sides

y= 3x3/3 +c

y=x3+c

Thus y=x3+c is a solution of differential equation dy/dx= 3x2

b) dy/dx= 3x2, y(0)=4

From the previous answer, y=x3+c

when x=0, y=4

4=03+c

c=4

rewriting the equation,y=x3+4

Hence y=x3+4 is solution of initial value, differential equation dy/dx= 3x2, y(0)=4.

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