Find the general solution of the given differential equation. x (dy/dx) - y = x² sin x?

Solution:

It is given that

x (dy/dx) - y = x² sin x -------> (1)

The standard form of linear differential equation is dy/dx+ P(x)y = f(x)

To convert the given equation (1) in the standard form, we divide both sides by x.

(1) becomes dy/dx -y/x = x sinx

⇒  dy/dx -(1/x)y = x sinx ------->(2)

Comparing with the standard form, we get P = 1/x

The integrating factor is e ^{∫P. dx}

IF = e ^{∫1/x. dx}

=e ^-ln|x|

=|x^-1| = |1/x|

= 1/x, if x > 0

Multiply IF with the equation (2), we get 

1/x. dy/dx -(1/x²)y = sinx

From this, any possible solution exists on either (-∞, 0) or (0, ∞)

It can be written as

1/x dy/dx - y/x² = sin x

By condensing the left side as a derivative of a product and by integrating both sides and solving for y

d/dx [y/x] = sin x

y/x = ∫ sin x dx

So we get

y/x = - cos x +C

y = Cx - x cos x

Therefore, the general solution is y = Cx - x cos x.

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Find the general solution of the given differential equation. x (dy/dx) - y = x² sin x?

Summary:

The general solution of the given differential equation x (dy/dx) - y = x² sin x is y = Cx - x cos x.