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प्रश्न
For the differential equation, find the general solution:
`dy/dx + 2y = sin x`
उत्तर
The given equation is `dy/dx + 2y = sin x.` ....(1)
Which is a linear equation of type `dy/dx + Py = Q.`
Here P = 2 and Q = sin x.
∴ `I.F. = e^(int Pdx) = e^(int2 dx) = e^(2x)`
∴ The solution is `y .(I.F.) =int Q. (I. F.) dx + C`
`y. e^(2x) = int e^(2x) sin x dx + C = I + C` ....(2)
Now, `I = int e^(2x) sin x dx`
`= e^(2x) (- cos x) - int 2e^(2x) (- cos x) dx` ...[Integrating by part]
`= -e^(2x) cos x + 2 int e^(2x) cosx dx`
Again integrating parts,
`I = - e^(2x) cos x + 2 [e^(2x) sin x - int e^(2x) * 2 sin x dx]`
`= I = - e^(2x) cos x + 2e^(2x) sin x - 4I`
⇒ 5I = e2x (2 sin x - cos x)
⇒`I = (e^(2x))/5 (2 sin x - cos x)` ....(3)
Substituting the value of (3) in (2), we get
`y. e^(2x) = 1/5 e^(2x) (2 sin x - cos x) + C`
⇒ `y = 1/5 (2 sin x - cos x) + Ce^(-2x),`
Which is the required solution.
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