Solve the following initial value problems: (xy-y^2)dx-x^2dy=0,y(… - Vidyadip

Question

Solve the following initial value problems:
$(\text{xy}-\text{y}^2)\text{dx}-\text{x}^2\text{dy}=0,\text{y}(1)=1$

✓

Answer

$(\text{xy}-\text{y}^2)\text{dx}-\text{x}^2\text{dy}=0,\text{y}(1)=1$
It is a homogeneous equation
Put y = vx
and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So,
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{xvx}-\text{v}^2\text{x}^2}{\text{x}^2}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\text{v}-\text{v}^2-\text{v}$
$\text{x}\frac{\text{dv}}{\text{dx}}=-\text{v}^2$
$-\int\frac{1}{\text{v}^2}\text{dv}=\int\frac{\text{dx}}{\text{x}}$
$-\Big(-\frac{1}{\text{v}}\Big)=\log|\text{x}|+\text{C}$
$\frac{\text{x}}{\text{y}}=\log|\text{x}|+\text{C}\ \dots(\text{i})$
Put y = 1, x = 1
1 = C
Using equation (1),
$\text{x}=\text{y}\big[\log|\text{x}|+1\big]$
$\text{y}=\frac{\text{x}}{\big[\log|\text{x}|+1\big]}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

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.

The bag $A$ contains $8$ white and $7$ black balls while the bag $B$ contains $5$ white and $4$ black balls. One ball is randomly picked up from the bag $A$ and mixed up with the balls in bag $B$. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

Find the image of the point (0, 0, 0) in the plane 3x + 4y - 6z + 1 = 0.

Find the angle of intersecting of the following curves:
$\text{x}^2+\text{y}^2-4\text{x}-1=0\text{ and }\text{x}^2+\text{y}^2-2\text{y}-9=0$

Solve the following determinant equations:
$\begin{vmatrix}\text{x}+1&3&5\\2&\text{x}+2&5\\2&3&\text{x}+4\end{vmatrix}=0$

Solve the follwing system of equations by matrix method: $x + y + z = 3 , 2x - y + z = -1 , 2x + y - 3z = -9$

Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-1}{3}=\frac{\text{y}-1}{-1}=\frac{\text{z}+1}{0}$ and $\frac{\text{x}-4}{2}=\frac{\text{y}-0}{0}=\frac{\text{z}+1}{3}$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$4\text{x}+3\text{y}=\log\big(4\text{x}-3\text{y}\big)$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=(\text{x}+\text{y}+1)^{2}$

Find the value of $\tan^{-1}\Big(\frac{\text{x}}{\text{y}}\Big)-\tan^{-1}\Big(\frac{\text{x-y}}{\text{x+y}}\Big)$