Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 13226 by chux last updated on 16/May/17

$$\mathrm{if}{\mathrm{x}}^3+{\mathrm{y}}^3=3\mathrm{axy},\mathrm{find}\mathrm{dy}/\mathrm{dx}\mathrm{in}\mathrm{terms}$$$$\mathrm{of}\mathrm{x}\mathrm{and}\mathrm{y}\mathrm{and}\mathrm{prove}\mathrm{that}\mathrm{dy}/\mathrm{dx}$$$$\mathrm{cannot}\mathrm{be}\mathrm{equal}\mathrm{to}-1\mathrm{for}\mathrm{finite}$$$$\mathrm{values}\mathrm{of}\mathrm{x}\mathrm{and}\mathrm{y}\mathrm{except}\mathrm{x}=\mathrm{y}.$$$$$$$$$$$$\mathrm{please}\mathrm{help}$$

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/May/17

$$3x^2+3y^2\frac{dy}{dx}=3ay+3ax\frac{dy}{dx}(\frac{dy}{dx}=y^{{}^{\prime }})$$$$\Rightarrow (3y^2-3ax)y^{{}^{\prime }}=3(ay-x^2)\Rightarrow$$$$y^{{}^{\prime }}=\frac{ay-x^2}{y^2-ax}.◼$$$$if:(y^{{}^{\prime }}=-1)\Rightarrow ay-x^2=ax-y^2\Rightarrow$$$$(y-x)(y+x)+a(y-x)=0\Rightarrow (y-x)(y+x+a)=0\Rightarrow$$$$\Rightarrow \{\begin{array}{l}y-x=0\Rightarrow y=x\\ y+x+a=0\Rightarrow y=-(x+a).◼\end{array}$$

Answered by ajfour last updated on 16/May/17

$$x^3+y^3=3axy......\left(i\right)$$$$3x^2+3y^2\frac{dy}{dx}=3ay+3ax\frac{dy}{dx}$$$$\frac{dy}{dx}=\frac{ay-x^2}{y^2-ax}$$$$\frac{dy}{dx}+1=\frac{ay-x^2+y^2-ax}{y^2-ax}$$$$=\frac{(y-x)(x+y+a)}{y^2-ax}$$$$\Rightarrow \frac{dy}{dx}+1=0onlywhen$$$$x=y,or(x+y+a)=0$$$${(x+y)}^3=x^3+y^3+3xy(x+y)...\left(ii\right)$$$$x^3+y^3=3axy...\left(i\right)$$$$\left(i\right)+\left(ii\right):$$$${(x+y)}^3=3xy(x+y+a)$$$$\Rightarrow if(x+y+a)=0,x+y=0$$$$sofor(x+y+a)=0conditionis$$$$x+y=0andeven\mathit{a}=0$$$$thereforeifa\ne 0,x+y+a\ne 0$$$$\Rightarrow \frac{dy}{dx}+1=0or\frac{dy}{dx}=-1only$$$$forx=y.\left(grantedtheyarefinite\right)$$

Commented by chux last updated on 17/May/17

$$\mathrm{thanks}\mathrm{alot}.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com