Question

If \[{{y}^{\dfrac{1}{m}}}+{{y}^{\dfrac{-1}{m}}}=2x\] then prove that \[\left( {{x}^{2}}-1 \right){{y}_{3}}+3x{{y}_{2}}+\left( 1-{{m}^{2}} \right){{y}_{1}}=0\].

Answer 1

Hint: Here ${{y}_{1}},{{y}_{2}},{{y}_{3}}$ are the first, second and third derivatives. First convert the given expression into simpler $y$ and $x$ terms and then start differentiating.

The given expression is,
\[{{y}^{\dfrac{1}{m}}}+{{y}^{\dfrac{-1}{m}}}=2x\]
This can be re-written as,
\[\Rightarrow {{y}^{\dfrac{1}{m}}}+\dfrac{1}{{{y}^{\dfrac{1}{m}}}}=2x\]
Now taking the LCM and solving, we get
\[\Rightarrow \dfrac{{{\left( {{y}^{\dfrac{1}{m}}} \right)}^{2}}+1}{{{y}^{\dfrac{1}{m}}}}=2x\]
On Cross multiplying, we get
\[\Rightarrow {{\left( {{y}^{\dfrac{1}{m}}} \right)}^{2}}+1=2x{{y}^{\dfrac{1}{m}}}\]
\[\Rightarrow {{\left( {{y}^{\dfrac{1}{m}}} \right)}^{2}}-2x{{y}^{\dfrac{1}{m}}}+1=0\]
Let \[{{y}^{\dfrac{1}{m}}}=z\], then above equation becomes
\[\Rightarrow {{z}^{2}}-2xz+1=0\]
This is a quadratic equation. The general quadratic equation is $a{{x}^{2}}+bx+c=0$, comparing the above equation with this we get
$a=1,b=-2x,c=1$
The root of this quadratic equation is given by
$z=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the corresponding values, we get
\[z=\dfrac{-(-2x)\pm \sqrt{{{\left( -2x \right)}^{2}}-4\left( 1 \right)\left( 1 \right)}}{2\left( 1 \right)}\]
\[\Rightarrow z=\dfrac{2x\pm \sqrt{4{{x}^{2}}-4}}{2}\]
Taking $4$ common under the root and taking out, we get
\[\Rightarrow z=\dfrac{2x\pm 2\sqrt{{{x}^{2}}-1}}{2}\]
Taking out $2$ common, we get
\[\therefore z=x\pm \sqrt{{{x}^{2}}-1}\]
Substituting back the value of \[z\], we get
\[{{y}^{\dfrac{1}{m}}}=z=x\pm \sqrt{{{x}^{2}}-1}\]
\[\Rightarrow {{y}^{\dfrac{1}{m}}}=x\pm \sqrt{{{x}^{2}}-1}\]
Powering both sides by $m$, we get
\[{{\left( {{y}^{\dfrac{1}{m}}} \right)}^{m}}={{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m}}\]
\[\therefore y={{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m}}.........(i)\]
Now differentiating the above expression with respect to \[x\], we get
\[{{y}_{1}}=\dfrac{d}{dx}{{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m}}\]
We know, $\dfrac{d}{dx}({{y}^{n}})=n{{y}^{n-1}}\dfrac{d}{dx}(y)$ , so the above equation becomes,
\[{{y}_{1}}=\dfrac{dy}{dx}=m{{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m-1}}\dfrac{d}{dx}\left( x\pm \sqrt{{{x}^{2}}-1} \right)\]
Applying the sum rule of differentiation, we get
\[{{y}_{1}}=\dfrac{dy}{dx}=m{{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m-1}}\left[ \dfrac{d}{dx}\left( x \right)\pm \dfrac{d}{dx}\left( \sqrt{{{x}^{2}}-1} \right) \right]\]
\[\Rightarrow {{y}_{1}}=m{{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m-1}}\times \left( 1\pm \dfrac{1}{2}\times \dfrac{1}{\sqrt{{{x}^{2}}-1}}\times 2x \right)\]
Taking the cancelling the lie terms and taking the LCM, we get
\[\Rightarrow {{y}_{1}}=m{{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m-1}}\times \left( \dfrac{\sqrt{{{x}^{2}}-1}\pm x}{\sqrt{{{x}^{2}}-1}} \right)\]
Cross multiplying, we get
\[\Rightarrow {{y}_{1}}\sqrt{{{x}^{2}}-1}=m{{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m-1}}\times \left( \sqrt{{{x}^{2}}-1}\pm x \right)\]
\[\Rightarrow {{y}_{1}}\sqrt{{{x}^{2}}-1}=m{{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m}}\]
Now substituting \[y={{\left( x\pm \sqrt{{{x}^{2}}-1} \right)}^{m}}\] from equation (i) in the above equation, we get
\[\Rightarrow {{y}_{1}}\sqrt{{{x}^{2}}-1}=my\]
Squaring on both sides, we get
\[\Rightarrow {{\left( {{y}_{1}}\sqrt{{{x}^{2}}-1} \right)}^{2}}={{\left( my \right)}^{2}}\]
\[\Rightarrow {{y}_{1}}^{2}\left( {{x}^{2}}-1 \right)={{m}^{2}}{{y}^{2}}\]
Now differentiating the above equation with respect to $'x'$ , we get
\[\Rightarrow \dfrac{d}{dx}\left( {{y}_{1}}^{2}\left( {{x}^{2}}-1 \right) \right)=\dfrac{d}{dx}\left( {{m}^{2}}{{y}^{2}} \right)\]
Now we know, \[\dfrac{d}{dx}\left( u.v \right)=u\dfrac{d}{dx}v+v\dfrac{d}{dx}u\], applying this formula, we get
\[\Rightarrow {{y}_{1}}^{2}\dfrac{d}{dx}\left( {{x}^{2}}-1 \right)+\left( {{x}^{2}}-1 \right)\dfrac{d}{dx}\left( {{y}_{1}}^{2} \right)={{m}^{2}}\dfrac{d}{dx}\left( {{y}^{2}} \right)\]
\[\Rightarrow {{y}_{1}}^{2}2x+\left( {{x}^{2}}-1 \right)2{{y}_{1}}\dfrac{d}{dx}\left( {{y}_{1}} \right)={{m}^{2}}2y\dfrac{d}{dx}(y)\]
Now we know, ${{y}_{1}}=\dfrac{dy}{dx},{{y}_{2}}=\dfrac{d{{y}_{1}}}{dx}$ , so the above equation becomes,
\[\Rightarrow {{y}_{1}}^{2}2x+\left( {{x}^{2}}-1 \right)2{{y}_{1}}{{y}_{2}}={{m}^{2}}2y{{y}_{1}}\]
\[\Rightarrow 2{{y}_{1}}\left( {{y}_{1}}x+\left( {{x}^{2}}-1 \right){{y}_{2}} \right)={{m}^{2}}2y{{y}_{1}}\]
Dividing throughout by $'2{{y}_{1}}'$ , we get
\[\Rightarrow {{y}_{1}}x+\left( {{x}^{2}}-1 \right){{y}_{2}}={{m}^{2}}y\]
Now again we will differentiate the above equation with respect to $'x'$, we get
\[\begin{align}
  & \Rightarrow \dfrac{d}{dx}\left( {{y}_{1}}x+\left( {{x}^{2}}-1 \right){{y}_{2}} \right)=\dfrac{d}{dx}\left( {{m}^{2}}y \right) \\
 & \Rightarrow \dfrac{d}{dx}\left( {{y}_{1}}x \right)+\dfrac{d}{dx}\left( \left( {{x}^{2}}-1 \right){{y}_{2}} \right)={{m}^{2}}\dfrac{d}{dx}\left( y \right) \\
\end{align}\]
Now we know, \[\dfrac{d}{dx}\left( u.v \right)=u\dfrac{d}{dx}v+v\dfrac{d}{dx}u\], applying this formula, we get
\[\Rightarrow {{y}_{1}}\dfrac{d}{dx}\left( x \right)+x\dfrac{d}{dx}\left( {{y}_{1}} \right)+\left( {{x}^{2}}-1 \right)\dfrac{d}{dx}\left( {{y}_{2}} \right)+{{y}_{2}}\dfrac{d}{dx}\left( {{x}^{2}}-1 \right)={{m}^{2}}\dfrac{d}{dx}\left( y \right)\]
Now we know, ${{y}_{1}}=\dfrac{dy}{dx},{{y}_{2}}=\dfrac{d{{y}_{1}}}{dx}$ , so the above equation becomes,
\[\Rightarrow {{y}_{1}}(1)+x{{y}_{2}}+({{x}^{2}}-1){{y}_{3}}+{{y}_{2}}2x={{m}^{2}}{{y}_{1}}\]
\[\Rightarrow {{y}_{1}}+x{{y}_{2}}+({{x}^{2}}-1){{y}_{3}}+{{y}_{2}}2x-{{m}^{2}}{{y}_{1}}=0\]
On regrouping, we get
\[\Rightarrow ({{x}^{2}}-1){{y}_{3}}+3x{{y}_{2}}+\left( 1-{{m}^{2}} \right){{y}_{1}}=0\]
Hence proved.

Note: In the given question we are asked to prove \[\left( {{x}^{2}}-1 \right){{y}_{3}}+3x{{y}_{2}}+\left( 1-{{m}^{2}} \right){{y}_{1}}=0\]
We should not confuse with the \[{{y}_{1}},{{y}_{2}}\,and\,{{y}_{3}}\]. \[{{y}_{1}},{{y}_{2}}\, and \,{{y}_{3}}\] are 1st, 2nd, and 3rd derivative of the given function.
If we directly apply differentiation to the given expression, it becomes lengthy and complicated.