Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Solve the following differential equation:
$\sqrt{\text{1 + x}^{2}+\text{y}^{2}+\text{x}^{2}\text{y}^{2}}+\text{xy}\frac{\text{dy}}{\text{dx}}=0.$
✓

Answer

Given differential equation can be written as
$\sqrt{\text{(1 + x}^{2})}\sqrt{(\text{1 + y}^{2})}+\text{xy}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\frac{\text{y}}{\sqrt{\text{1 + y}^{2}}}\text{ dy}=-\frac{\sqrt{\text{1 + x}^{2}}}{\text{x}}\text{dx}$
Integrating both sides, we get
$\sqrt{\text{1 + y}^{2}}=-\int\frac{\sqrt{\text{1 + x}^{2}}}{\text{x}^{2}}\cdot\text{x dx}=-\int\frac{\text{t}^{2}\text{ dt}}{\text{t}^{2}-1}\text{where }(1+\text{x}^{2})=\text{t}^{2}$
$\Rightarrow\sqrt{\text{1 + y}^{2}}=-\int\Bigg(1+\frac{1}{\text{t}^{2}-1}\Bigg)\text{dt}=-\text{t}-\frac{1}{2}\log\frac{\text{t - 1}}{\text{t + 1}}\text{c}$
$=-\sqrt{\text{1 + x}^{2}}-\frac{1}{2}\log\Bigg|\frac{\sqrt{\text{1 + x}^{2}}-1}{\sqrt{\text{1 + x}^{2}}+1}\Bigg|+\text{c}$
OR $\sqrt{\text{1 + y}^{2}}+\sqrt{\text{1 + x}^{2}}-\frac{1}{2}\log\Bigg|\frac{\sqrt{\text{1 + x}^{2}}-1}{\sqrt{\text{1 + x}^{2}}+1}\Bigg|=\text{c}$.

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 "5 Marks Questions" 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

Solve the following systems of homogeneous linear equations by matrix method:
2x + 3y - z = 0
x - y - 2z = 0
3x + y + 3z = 0
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{x}-2\sin\text{x}$
→
Find the maximum area of an isosceles $\triangle $ inscribed in the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with its vertex at one end of the major axis
→
Solve the following differential equation
$\sin\Big(\frac{\text{dy}}{\text{dx}}\Big)=\text{K};\text{y}(0)=1$
→
Show that of all the rectangles inscribed in a given fixed circle, the square has maximum area.
→
Differentiate $\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$ with respect to $\sqrt{1-4\text{x}^2},$ if:
$\text{x}\in\Big(-\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2\sqrt{2}}}\Big)$
→
From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.
→
Evaluate the following integrals as limit of sum:
$\int\limits^3_1(3\text{x}-2)\text{dx}$
→
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-3\frac{\text{dy}}{\text{dx}}+2\text{y}=0,\text{y}(0)=1,\text{y}(0)=3$
Function $\text{y}=\text{e}^\text{x}+\text{e}^{2\text{x}}$
→
By computing the shortest distance determine whether the following pairs of lines intersect or not:
$\vec{\text{r}}=\big(\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}\big)+\mu\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$
→