`f''(x) = x^(-3/2), f'(4) = 2, f(0) = 0` Find the particular solution that satisfies the differential equation.

Textbook Question

Chapter 4, 4.1 - Problem 41 - Calculus of a Single Variable (10th Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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`f''(x)=x^(-3/2)`

`f'(x)=intx^(-3/2)dx`

`f'(x)=x^(-3/2+1)/(-3/2+1)+C_1`

`f'(x)=-2x^(-1/2)+C_1`

`f'(x)=-2/sqrt(x)+C_1`

Now , solve for C_1 , given f'(4)=2 

`2=-2/sqrt(4)+C_1`

`2=-1+C_1`

`C_1=3`

`:.f'(x)=-2/sqrt(x)+3`

`f(x)=int(-2/sqrt(x)+3)dx`

`f(x)=-2((x^(-1/2+1))/(-1/2+1))+3x+C_2`

`f(x)=-4sqrt(x)+3x+C_2`

Now let's find out C_2 , given f(0)=0

`0=-4sqrt(0)+3(0)+C_2`

`C_2=0`

`:.f(x)=-4sqrt(x)+3x`

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