Let y = y(x) be the solution of the differential equation
\[{{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x\] such that y(0) = 7. Then y(\[\pi\]) is equal to :
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Official solution
\[{{dy} \over {dx}} - 2xy = 2(2\sin x - 5)x - 2\cos x\]
IF = e−x2
So, \[y.{e^{ - {x^2}}} = \int {{e^{ - {x^2}}}(2x(2\sin x - 5) - 2\cos x)dx} \]
\[ \Rightarrow y.{e^{ - {x^2}}} = {e^{ - {x^2}}}(5 - 2\sin x) + c\]
\[ \Rightarrow y = 5 - 2\sin x + c.{e^{{x^2}}}\]
Given at x = 0, y = 7
\[\Rightarrow\] 7 = 5 + c \[\Rightarrow\] c = 2
So, \[y = 5 - 2\sin x + 2{e^{{x^2}}}\]
Now, at x = \[\pi\],
y = 5 + 2ex2