Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
In each of the verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
$\text{xy} = \text{log}\ \text{y} + \text{C} \ \ : \ \text{y}'=\frac{\text{y}^2}{1-\text{xy}}\ (\text{xy} \neq1)$
✓

Answer

Given: xy = log y + C ....(i)

To prove: y given by eq. (i) is a solution of differential equation $\text{y}'=\frac{\text{y}^2}{1-\text{xy}} \ ...(\text{ii})$

Proof: Differentiating both sides of eq. (i) w.r.t x, we have

$\text{xy}'+\text{y}(1)=\frac{1}{\text{y}}\text{y}'+0 \ $ $\Rightarrow \ \text{xy}'-\frac{\text{y}'}{\text{y}}=-\text{y} $ $ \Rightarrow \ \text{y}'\bigg(\text{x}-\frac{\text{1}}{\text{y}}\bigg)=-\text{y}$

$\Rightarrow \ \text{y}'\bigg(\frac{\text{xy}-1}{\text{y}}\bigg)=-\text{y}\ $ $\Rightarrow \ \text{y}'(\text{xy}-1)=-\text{y}^2$ $ \Rightarrow \ \text{y}'=\frac{-\text{y}^2}{\text{xy}-1}$

$\Rightarrow \ \ \text{y}'=\frac{-\text{y}^2}{-(1-\text{xy})}=\frac{\text{y}^2}{1-\text{xy}}$

Hence, Function (implicit) given by eq. (i) is a solution of ${\text{y}}'=\frac{\text{y}^2}{1-\text{xy}}.$

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 "3 Marks" from DIFFERENTIAL EQUATIONS→DIFFERENTIAL EQUATIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the integral of the function sin4 x
→
Let * be a binary operation on Q - {-1} defined by a * b = a + b + ab for all a, b ∈ Q - {-1}. Then,
Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element.
→
Verify that y = A cos x - b sin x is a solution of the differential equation.

$\frac{\text{d}^{2} \text{y}}{\text{dx}^{2}}+\text{y}=0.$

→
Find the angle between the line $\vec{\text{r}}=(2\hat{\text{i}}+3\hat{\text{j}}+9\hat{\text{k}})+\lambda(2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})$ and the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=5$ 
→
Find a vector perpendicular to vectors $\overrightarrow{ a }=\hat{ i }+2 \hat{ j }-3 \hat{ k }$ and $\overrightarrow{ b }=3 \hat{ i }-\hat{ j }+2 \hat{ k }$, whose magnitude is $\sqrt{171}$.
→
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$ $\text{y}=\pm\sqrt{\text{a}^2-\text{x}^2}$
→
A bag contains 1 white and 6 red balls, and a second bag contains 4 white and 3 red balls. One of the bags is picked up at random and a ball is randomly drawn from it, and is found to be white in colour. Find the probability that the drawn ball was from the first bag.
→
Evalute the following integrals:
$\int\frac{\cos\text{x}}{\cos(\text{x}-\text{a})}\text{dx}$
→
Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=\sqrt{\text{x}^3}\ \text{at}\text{ x}=4$
→
Write the composition table for the binary operation multiplication modulo 10 (×10) on the set S = {2, 4, 6, 8}.
→