The differential equation 2xy , dy = (x^2 + y | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given the differential equation: $2xy \, dy = (x^2 + y^2 + 1) dx$ Rearranging the terms: $2xy \, dy - y^2 dx = (x^2 + 1) dx$ Divide both sides by

Vedclass · Accepted Answer

$A$ family of rectangular hyperbola with centre on $x$-axis

Vedclass · Answer

(C) Given the differential equation: $2xy \, dy = (x^2 + y^2 + 1) dx$ Rearranging the terms: $2xy \, dy - y^2 dx = (x^2 + 1) dx$ Divide both sides by $x^2$: $\frac{2xy \, dy - y^2 dx}{x^2} = \frac{(x^2 + 1) dx}{x^2}$ This can be written as: $d\left(\frac{y^2}{x}\right) = (1 + x^{-2}) dx$ Integrating both sides: $\frac{y^2}{x} = x - \frac{1}{x} + C$ Multiplying by $x$: $y^2 = x^2 - 1 + Cx$ Rearranging to standard form: $x^2 - Cx - y^2 = 1$ Completing the square for $x$: $(x - \frac{C}{2})^2 - y^2 = 1 + \frac{C^2}{4}$ This equation represents a family of rectangular hyperbolas with the center at $(\frac{C}{2}, 0)$,which lies on the $x$-axis.

The differential equation $2xy \, dy = (x^2 + y^2 + 1) dx$ determines

Similar Questions

The family of curves in which the sub-tangent at any point to any curve is double the abscissa is given by

If surrounding air is kept at $20^{\circ} C$ and a body cools from $80^{\circ} C$ to $70^{\circ} C$ in $5$ minutes,then the temperature of the body after $15$ minutes will be: (in $^{\circ} C$)

If the population grows at the rate of $8 \%$ per year,then the time taken for the population to be doubled,is (Given $\log 2 = 0 \cdot 6912$)

The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in $5$ hours,then in $25$ hours,its number would be ......... times the original.

The principal increases continuously in a newly opened bank at the rate of $10 \%$ per year. An amount of Rs. $2000$ is deposited with this bank. How much will it become after $5$ years? $(e^{0.5} = 1.648)$

Vedclass Test Series

Exam Paper Generator

Online Exam Module