`f''(x) = sin(x), f'(0) = 1, f(0) = 6` Find the particular solution that satisfies the differential equation.

Textbook Question

Chapter 4, 4.1 - Problem 42 - Calculus of a Single Variable (10th Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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`f''(x)=sin(x)`

`f'(x)=intsin(x)dx`

`f'(x)=-cos(x)+C_1`

Let's find constant C_1 , given f'(0)=1

`f'(0)=1=-cos(0)+C_1`  

`1=-1+C_1`

`C_1=2`

`:.f'(x)=-cos(x)+2`

`f(x)=int(-cos(x)+2)dx`

`f(x)=-sin(x)+2x+C_2`

Now let's find constant C_2 . given f(0)=6

`f(0)=6=-sin(0)+2(0)+C_2`

`6=-0+0+C_2`

`C_2=6`

`:.f(x)=-sin(x)+2x+6`

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