   Chapter 5.1, Problem 54E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding a Particular Solution In Exercises 51–54, find a function f that satisfies the differential equation and the initial conditions. f " ( x ) = x − 3 / 2 ,   f ' ( 1 ) = 2 ,   f ( 9 ) = − 4

To determine

To calculate: The particular solution of differential equation f''(x)=x3/2 with initial condition f'(1)=2 and f(9)=4.

Explanation

Given Information:

The differential equation is f''(x)=x3/2, and the initial condition are f'(1)=2 and f(9)=4.

Formula used:

The simple power rule of integration xndx=xn+1n+1+C.

Calculation:

Consider the differential equation, f''(x)=x3/2.

Integrate the provided differential equation, use the constant rule of integration xndx=xn+1n+1+C.

f''(x)dx=x3/2dxf'(x)=[x32+132+1]+C1=x1212+C1=2x12+C1

Now, from the initial condition of the differential function f'(x).

Substitute 1 for x in differential equation f'(x)=2x12+C1.

f'(1)=2(1)12+C1=2+C1

Substitute 2 for f'(8) in above differential equation to get the value of constant C1.

2=2+C1C1=2+2=4

Substitute 4 for C1 in the differential function f'(x)=2x12+C1.

f'(x)=2x12+4

Integrate the above differential equation f'(x)=2x12+4, use the simple power rule of integration xndx=xn+1n+1+C

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