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Question Number 90952 by jagoll last updated on 27/Apr/20

$$$$ $$solvediffequation$$ $$x^2{y}^{″}+{xy}^{\prime }+y=5x^{2}x>0$$

Answered by MWSuSon last updated on 27/Apr/20

$$letx=e^z$$ $$x^2{y}^{″}=D(D-1)y$$ $${xy}^{\prime }=Dy$$ $$(D^2+1)y=5e^{2z}$$ $$Auxillaryequation$$ $$m^2+1=0$$ $$m=\pm i$$ $$y_c=C_{1}Cos\left(z\right)+C_{2}Sin\left(z\right)$$ $$y_p=5\frac{1}{D^2+1}{e}^{2z}$$ $$y_p=5\frac{1}{5}{e}^{2z}$$ $$y_p=e^{2z}$$ $$y=C_{1}Cos\left({log}_{e}x\right)+C_{2}Sin\left({log}_{e}x\right)+x^2$$ $$wecanseethatxcannotbe-ve.$$

Commented byjagoll last updated on 27/Apr/20

$$sir.wheredoesformD(D-1)y?$$

Commented byMWSuSon last updated on 27/Apr/20

$$yourquestionisaCauchyEuler$$ $$D.E,whilederivingawaytosolve$$ $$theequationweendeduphaving$$ $$x\frac{dy}{dx}=Dy,x^2\frac{d^{2}y}{{dx}^2}=D(D-1)y,x^3\frac{d^{3}y}{{dx}^3}=D(D-1)(D-2)y...$$ $$D\equiv \frac{d}{dz}.allthisisgottenfrommaking$$ $$x=e^z.$$

Commented byjagoll last updated on 27/Apr/20

$$ziscomplexnumbersir?$$

Commented byMWSuSon last updated on 27/Apr/20

$$noitisnotacomplexnumber.$$

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