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Question Number 193733 by Tawa11 last updated on 18/Jun/23

$$\mathrm{Find}\mathrm{the}\mathrm{ordinary}\mathrm{differential}\mathrm{equation}$$$$\mathrm{satisfy}\mathrm{by}:$$$$\mathrm{y}={\mathrm{x}}^{\mathrm{n}}(\mathrm{A}+\mathrm{Blogx})$$

Answered by Rajpurohith last updated on 19/Jun/23

$$y{}^{\prime }=x^n\left(\frac{B}{x}\right)+{nx}^{n-1}(A+Blog\left(x\right))$$$$={Bx}^{n-1}+\left(\frac{n}{x}\right)y$$$$\Rightarrow xy{}^{\prime }={Bx}^n+ny....\left(1\right)$$$$differentiateagain$$$${xy}^{″}+y^{{}^{\prime }}={Bx}^{n-1}+{ny}^{\prime }$$$$\Rightarrow x^2{y}^{″}+{xy}^{\prime }={Bx}^n+{nxy}^{\prime }....\left(2\right)$$$$subtract\left(1\right)from\left(2\right)$$$$x^2{y}^{″}+{xy}^{\prime }-{xy}^{\prime }={Bx}^n+{nxy}^{\prime }-{Bx}^n-ny$$$$\Rightarrow x^2{y}^{″}-{nxy}^{\prime }+ny=0$$$$istherequiredODE.$$$$$$

Commented by Tawa11 last updated on 19/Jun/23

$$\mathrm{I}\mathrm{appreciate}\mathrm{sir}.$$$$\mathrm{God}\mathrm{bless}\mathrm{you}.$$

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