Question

Solve (D^2+1)y=Cos 2x

Answer 1

Answer:

Step-by-step explanation:

Hope it helps you

Answer 2

The general solution of (D² + 1)y = Cos(2x) is c1 cos(x) + c2 sin(x)  + x/2 sin(2x).

Given,

(D² + 1)y = Cos(2x)

To Find,

solution of given equation

Solution,

We can solve this problem using a simple method.

The given equation is (D² + 1)y = Cos(2x).

This is a second-order differential equation.

Here, the auxiliary equation (AE) can be written as:

⇒ D² + 1 = 0

⇒ D² = -1

⇒ D = ± i

The complementary function (CF) is given by:

⇒ CF = c1 cos(x) + c2 sin(x)

where c1 and c2 are arbitrary constants.

Now, the particular integral PI is given by the following:

$\implies PI = \frac{1}{D^2 + 1} \hspace{0.1cm} cos (2x)$

Put D² = -1²

$\implies PI = \frac{1}{-1^2 + 1} \hspace{0.1cm} cos (2x)\\\\\implies PI = \frac{1}{0} \hspace{0.1cm} cos (2x)$

Here, our method fails.

Hence, PI could be written as:

$\implies PI = \frac{1}{D^2 + 1} \hspace{0.1cm} cos (2x)\\\\\implies PI = x \int cos (2x) \hspace{0.1cm} dx\\\\\implies PI = x \hspace{0.1cm} \frac{sin(2x)}{2}\\\\\implies PI = \frac{x}{2} \hspace{0.1cm} sin(2x)$

Therefore, the general solution (GS) of the given differential equation is given by:

⇒ GS = CF + PI

⇒ GS = c1 cos(x) + c2 sin(x)  + x/2 sin(2x)

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