The differential equation x frac{dy}{dx} = 2y | vedclass

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

Question

(B) Given the differential equation: $x \frac{dy}{dx} = 2y$. Separating the variables,we get: $\frac{dy}{y} = 2 \frac{dx}{x}$. Integrating both sides:

Vedclass · Accepted Answer

a family of parabolas with vertex at the origin and axis along the positive $Y$-axis.

Vedclass · Answer

(B) Given the differential equation: $x \frac{dy}{dx} = 2y$. Separating the variables,we get: $\frac{dy}{y} = 2 \frac{dx}{x}$. Integrating both sides: $\int \frac{dy}{y} = 2 \int \frac{dx}{x}$. This gives: $\ln|y| = 2 \ln|x| + C$. Using properties of logarithms: $\ln|y| = \ln|x^2| + C$. Taking the exponential of both sides: $y = e^{\ln|x^2| + C} = e^C \cdot x^2$. Let $e^C = k$,where $k$ is an arbitrary constant. Thus,$y = kx^2$. This equation represents a family of parabolas with the vertex at the origin $(0,0)$ and the axis along the $Y$-axis.

The differential equation $x \frac{dy}{dx} = 2y$ represents ....

Similar Questions

The population $P = P(t)$ at time $t$ of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$,then the time at which the population becomes zero is:

The solution of the differential equation $y \frac{dy}{dx} + x = k$ represents . . . . . . .

Any curve passes through the point $(3, -4)$. If the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2y}{x}$,then the equation of the curve is . . . . . . .

In a culture,the bacteria count is $1,00,000$ initially. The number increases by $10 \%$ in the first $2$ hours. In how many hours will the count reach $2,00,000$,if the rate of growth of bacteria is proportional to the number present?

The rate of growth of bacteria is proportional to the bacteria present. If it is found that the number doubles in $3$ hours,then the number of times the bacteria are increased in $6$ hours is

Vedclass Test Series

Exam Paper Generator

Online Exam Module