Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=2(y+2 \sin x-5) x-2 \cos x$ such that $y(0)=7$. Then $\mathrm{y}(\pi)$ is equal to :
Select the correct option:
A
$2 \mathrm{e}^{\pi 2}$
B
$e^{\pi^2+5}$
C
$3 e^{\pi^2}+5$
D
$7 e^{\pi^2}+5$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
dy/dx − 2xy = 2 (2sin x − 5)x − 2 cos x
IF = e^(−x²)
So, y·e^(−x²) = ∫ [e^(−x²)(2x)(2sin x − 5) − 2 cos x] dx
⇒ y e^(−x²) = e^(−x²)(5 − 2 sin x) + C
⇒ y = 5 − 2 sin x + C e^(x²)
Given at x = 0, y = 7
⇒ 7 = 5 + C ⇒ C = 2
So, y = 5 − 2 sin x + 2e^(x²)
Now at x = π,
y = 5 + 2e^(π²)
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