Question

12th board important question! ...

Differentiate it .​

Answer 1

Question :

Solve :

$\tt\sqrt{a+x}\dfrac{dy}{dx}+x=0$

Solution :

We have ,

$\sf\sqrt{a+x}\dfrac{dy}{dx}+x=0$

$\sf\implies\sqrt{a+x}\dfrac{dy}{dx}=-x$

$\sf\implies\dfrac{dy}{dx}=\dfrac{-x}{\sqrt{a+x}}$

$\sf\implies\:dy=\dfrac{-x}{\sqrt{a+x}}dx$

Now integrate on both sides

$\sf\int\:dy=-\int\dfrac{x}{\sqrt{a+x}}dx$

$\sf\int\:dy=-[\int\dfrac{a+x-a}{\sqrt{a+x}}dx]$

$\sf\:y=-\int\dfrac{a+x}{\sqrt{a+x}}dx+\int\dfrac{a}{\sqrt{a+x}}dx$

$\sf\:y=-\int(a+x)^{\frac{1}{2}}dx+\int\:a(a+x)^{\frac{-1}{2}}dx$

We know that

$\sf\int\:x^n\:dx=\dfrac{x^{n+1}}{n+1}$

Then,

$\sf\:y=-\int(a+x)^{\frac{1}{2}}dx+\int\:a(a+x)^{\frac{-1}{2}}dx$

$\sf\:y=-\dfrac{(a+x)^{\frac{1}{2}+1}}{\frac{1}{2}+1}+a\dfrac{(a+x)^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+c$

$\sf\:y=-\dfrac{(a+x)^{\frac{3}{2}}}{\frac{3}{2}}+a\dfrac{(a+x)^{\frac{1}{2}}}{\frac{1}{2}}+c$

$\tt\:y=\dfrac{-2}{3}(a+x)^{\frac{3}{2}}+2a(a+x)^{\frac{1}{2}}+c$

It is the required solution!

Answer 2

Answer:

Question :

Solve :

$\tt\sqrt{a+x}\dfrac{dy}{dx}+x=0 </p><p>$

Solution :

We have ,

$\sf\sqrt{a+x}\dfrac{dy}{dx}+x=0$

$\sf\implies\sqrt \orange{a+x}\dfrac{dy}{dx}$

$\sf\implies\dfrac{dy}{dx}=\dfrac{-x}{\sqrt{a+x}}$

$\sf\implies\:dy=\dfrac{-x}{\sqrt{a+x}}$

Now integrate on both sides

$\sf\int\:dy=-\int\dfrac{x}{\sqrt{a+x}}$

$\sf\int\:dy=-[\int\dfrac{a+x-a}{\sqrt{a+x}}$

$\sf\:y=-\int\dfrac{a+x}{\sqrt{a+x}}dx+\int\dfrac{a}{\sqrt{a+x}}$

$\sf\:y=-\int(a+x)^{\frac{1}{2}}dx+\int\:a(a+x)^{\frac{-1}{2}}$

We know that

$\sf\int\:x^n\:dx=\dfrac \purple{x^{n+1}}{n+1}$

$\sf\:y=-\int(a+x)^{\frac{1}{2}}dx+\int\:a(a+x)^{\frac{-1}{2}}$

$\sf\:y=-\dfrac \pink{(a+x)^{\frac{1}{2}+1}}{\frac{1}{2}+1}+a\dfrac \gray{(a+x)^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+cy=− </p><p>$

$\sf\:y=-\dfrac \red{(a+x)^{\frac \green{3}{2}}}{\frac{3}{2}}+a\dfrac{(a+x)^{\frac{1}{2}}}{\frac{1}{2}}+cy=−$

$\tt\:y=\dfrac \blue{-2} \red{3} \gray(a+x)^{\frac{3}{2}}+2a(a+x)^{\frac{1}{2}}$

It is the required solution !!!