Find the general solution of the given differential equation xdydx−y=x2sin⁡x give the largest interval over which the genreral solution is defined.

xdydx−y=x2sin⁡x ⇒dydx−yx=xsin⁡x This is 1-st order linear in y so integrating Factor =e∫−1xdx =eln⁡(1x) =1x multiplying the given equation with I.F. we get 1xdydx−yx2=sin⁡x ⇒ddx(yx)=sin⁡x integrating both side we get ∫d(yx)=sin⁡x ⇒yx=−cos⁡x+C ⇒y=−xcos⁡x+Cx Since y is continuous on (−∞,∞) so longest inter val of existence is (−∞,∞) Since xcos⁡x and Cx both does not goes to 0 as x→∞ Hence then is no transient form.

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Linda Seales

Answered question

2021-12-13

Find the general solution of the given differential equation

x \frac{d y}{d x} - y = x^{2} \sin x

give the largest interval over which the genreral solution is defined.

Answer & Explanation

Bernard Lacey

Beginner2021-12-14Added 30 answers

x \frac{d y}{d x} - y = x^{2} \sin x

\Rightarrow \frac{d y}{d x} - \frac{y}{x} = x \sin x

This is 1-st order linear in y so integrating Factor

= e^{\int - \frac{1}{x} d x}

= e^{\ln (\frac{1}{x})}

= \frac{1}{x}

multiplying the given equation with I.F. we get

\frac{1}{x} \frac{d y}{d x} - \frac{y}{x^{2}} = \sin x

\Rightarrow \frac{d}{d x} (\frac{y}{x}) = \sin x

integrating both side we get

\int d (\frac{y}{x}) = \sin x

\Rightarrow \frac{y}{x} = - \cos x + C

\Rightarrow y = - x \cos x + C x

Since y is continuous on

(- \infty, \infty)

so longest inter val of existence is

(- \infty, \infty)

Since

x \cos x

and Cx both does not goes to 0 as

x \to \infty

Hence then is no transient form.

Ella Williams

Beginner2021-12-15Added 28 answers

This will be useful to me. Thanks to.

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