If x^ m y^ n = ( x + y)^ m + n , prove that d^ 2 y dx^ 2 = 0. - Vidyadip

Question

$\text{If } x^{\text{m }} \text{y}^{\text{n}} = ({x + \text{y)}^{\text{m + n}}}, \text{prove that} \frac{\text{d}^{2}\text{y}}{\text{d}x^{2}} = 0.$

✓

Answer

$\text{x}^{\text{m}} . \text{y}^{\text{n}} = \text{(x + y)}^{\text{m + n}}$
$\Rightarrow \text{m} \log \text{x + n} \log \text{y = (m + n)} \log \text{(x + y)}$
$\Rightarrow \frac{\text{m}}{\text{x}} + \frac{\text{n}}{\text{y}}.\frac{\text{dy}}{\text{dx}} = \frac{\text{m + n}}{\text{x + y}} \bigg(1 + \frac{\text{dy}}{\text{dx}}\bigg)$
$\Rightarrow \frac{\text{dy}}{\text{dx}} = \frac{\text{y}}{\text{x}} \text{ }\text{ }\text{ }\text{ }\text{ }\dots\text{(i)}$
$\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}} = \frac{\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}}{\text{x}^{2}} \text{ }\text{ }\text{ }\text{ }\text{ } \dots\text{(ii) }$
$= \frac{\text{x}{\frac{\text{y}}{\text{x}} -\text{y}}}{\text{x}^{2}} \text{ }\text{ }\text{ }\text{ }\text{ }\dots\text{using (i)}$
= 0

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following functions with respect to x:
$\log(3\text{x}+2)-\text{x}^2\log(2\text{x}-1)$

A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240kg of phosphoric acid, at least 270kg of potash and at most 310kg of chlorine.

Kg per bag
	Brand P	Brand P
Nitrogen	32	3.5
Phosphoric	1	2
Potash	3	1.5
Chlorine	1.5	2

If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{y}=0,\text{y}(0)=2,\text{y}'(0)=0$

Function $\text{y}=\text{e}^\text{x}+\text{e}^{-\text{x}}$

Find the shortest distance between the lines whose vector equations are:
$\vec{\text{r}}=\Big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\Big)+\lambda\Big(\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}\Big)$
$\text{and}\ \vec{\text{r}}=4\hat{\text{i}}+5\hat{\text{j}}+6\hat{\text{k}}+\mu\Big(2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}\Big)$

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman's time. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.

verify that $\text{y}^2=4\text{a}(\text{x}+\text{a})$ is a solution of the differential equation $\Big\{1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}=2\text{x}\frac{\text{dy}}{\text{dx}}.$

Evaluate the following intregals:
$\int\frac{\text{x}}{(\text{x}^2-\text{a}^2)(\text{x}^2-\text{b}^2)}\ \text{dx}$

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Prove that:
$\begin{vmatrix}-\text{bc}&\text{b}^2+\text{bc}&\text{c}^2+\text{bc}\\\text{a}^2+\text{ac}&-\text{ac}&\text{c}^2+\text{ac}\\\text{a}^2+\text{ab}&\text{b}^2+\text{ab}&-\text{ab}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})^3$