Solve the following differential equations: x 1-y^2 dx+y 1-x^2 dy… - Vidyadip

Question

Solve the following differential equations:

$\text{x}\sqrt{1-\text{y}^2}\text{dx}+\text{y}\sqrt{1-\text{x}^2}\text{dy}=0$

✓

Answer

We have,
$\text{x}\sqrt{1-\text{y}^2}\text{dx}+\text{y}\sqrt{1-\text{x}^2}\text{dy}=0$
$\Rightarrow\text{y}\sqrt{1-\text{x}^2}\text{dy}=-\text{x}\sqrt{1-\text{y}^2}\text{dx}$
$\Rightarrow\frac{\text{y}}{\sqrt{1-\text{y}^2}}\text{dy}=-\frac{\text{x}}{\sqrt{1-\text{x}^2}}\text{dx}$
Integrating both sides, we get
$\int\frac{\text{y}}{\sqrt{1-\text{y}^2}}\text{dy}=-\int\frac{\text{x}}{\sqrt{1-\text{x}^2}}\text{dx}$
Substituting $1-\text{y}^2=\text{t}$ and $1-\text{x}^2=\text{u},$ we get
$-2\text{y dy = dt}$ and $-2\text{x dy = du}$
$\therefore\frac{-1}{2}\int\frac{1}{\sqrt{\text{t}}}\text{dt}=\frac{1}{2}\int\frac{1}{\sqrt{\text{u}}}\text{du}$
$\Rightarrow-\text{t}^{\frac{1}{2}}=\text{u}^{\frac{1}{2}}+\text{K}$
$\Rightarrow\sqrt{1-\text{x}^2}+\sqrt{1-\text{y}^2}=-\text{K}$
$\Rightarrow\sqrt{1-\text{x}^2}+\sqrt{1-\text{y}^2}=\text{C}$ (where, C = K)
Hence, $\sqrt{1-\text{x}^2}+\sqrt{1-\text{y}^2}=\text{C}$ is the required solution.

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 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

For the following matrices verify the associativity of multiplication i.e., (AB) C = A(BC):
$\text{A}=\begin{bmatrix}1&2&0\\-1&0&1\end{bmatrix},\text{B}=\begin{bmatrix}1&0\\-1&2\\0&3\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1\\-1\end{bmatrix}$

Evalute the following integrals:
$\int\frac{1-\cot\text{x}}{1+\cot\text{x}}\text{dx}$

A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.

Evaluate the following integrals:

$\int\frac{1}{\sqrt{(\text{x}-\alpha)(\beta-\text{x})}}\text{ dx},(\beta>\alpha)$

Evaluate: $\int \frac{\sin x}{(1 - \cos x) ( 2 - cos x)} \text{dx}$

A particle moves along the curve $\text{y}=\big(\frac{2}{3}\big)\text{x}^3+1.$ Find the points on the curve at which the y-coordinate is changing twice as fast as the x-coordinate.

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\cos^2\text{x dx}$

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

Evaluate the following intregals:
$\int\frac{1}{3+2\sin\text{x}+\cos\text{x}}\ \text{dx}$

Differentiate $\sin^{-1}\Big(2\text{ax}\sqrt{1-\text{a}^2\text{x}^2}\Big)$ with respect to $\sqrt{1-\text{a}^2\text{x}^2},$ if $-\frac{1}{\sqrt{2}}<\text{ax}<\frac{1}{\sqrt{2}}$.