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Problem statement

Please see the step enclosed in yellow ring in the above image. My question is why we cannot apply Componendo-Dividendo so the integration becomes easy.

The correct answer to the question is (x - y)^2 = Cxy × e^(-y/x) but I do not get this answer by applying Componend-Dividendo.

Why am I not getting correct answer? Please help.

Here is what I did : enter image description here

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    I have included the picture. Please include your work. It's possible that there is something going wrong in one of your steps, but we can't know that if we don't know what they are. – 2017-02-06
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    $\dfrac{1+v}{1-v}$ does not equal $\dfrac{1+v+1-v}{1-v-1-v}$. My understanding is that componendo and dividendo apply to equations, not individual fractions. – 2017-02-06

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Edited: The issue is in the very first step. Componendo and dividendo are applicable to equations where both sides of the equation are rational expressions. In other words, if we know that $$\frac{a}{b}=\frac{c}{d}$$ where neither side is equal to $1$ then componendo and dividendo let us conclude that $$\frac{a+b}{a-b}=\frac{c+d}{c-d}.$$

They are not applicable to individual rational expressions. Indeed, if we could do as you are saying, then $$\frac32=\frac{3+2}{3-2}=\frac{5}{1},$$ which is clearly absurd.

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    I applied to the fraction alone. If you call (1 + v)/(1 - v) = f (v). Then it applied componendo-dividendo to f (v) then subtituted in that step. – 2017-02-06
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    Then componendo-dividendo seems to mean something different to you than it does to me. Please show your work, or I won't be able to help you. – 2017-02-06
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    please take a look. I have uploaded the image of what I did. – 2017-02-06
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    I have updated my answer accordingly. – 2017-02-06