How can I solve the differential equation y'=sin x−xsin x ?

Lina Neal 2022-09-12 Answered
How can I solve the differential equation y = sin x - x sin x ?
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Answers (1)

incibracy5x
Answered 2022-09-13 Author has 21 answers
y = d y d x = sin x - x sin x
To find y, we have to take the integral of y':
y = ( sin x - x sin x ) d x
y = sin x d x - x sin x d x = - cos x - I
I = x sin x d x
The argument of the integral is product of two functions. As such, we will use integration by parts:
u = x , and d v = sin x d x
d u = d x , and v = - cos x
u d v = u v - v d u
x sin x d x = - x cos x - - cos x d x =
- x cos x + cos x d x = - x cos x + sin x
I = - x cos x + sin x
Let's plug it in:
y = - cos x - ( - x cos x + sin x ) = - cos x + x cos x - sin x
y = ( x - 1 ) cos x - sin x + C
This is the solution to the differential equation in your problem statement.

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