The solution of the differential equation x^2 | vedclass

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

Question

(A) The given equation is a homogeneous differential equation: $\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2} = 1 + \frac{y}{x} + \left( \frac{y}{x} \rig

Vedclass · Accepted Answer

$	an^{-1} \left( \frac{y}{x} ight) = \log x + c$

Vedclass · Answer

(A) The given equation is a homogeneous differential equation: $\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2} = 1 + \frac{y}{x} + \left( \frac{y}{x} ight)^2$. Substitute $y = vx$,which implies $\frac{dy}{dx} = v + x \frac{dv}{dx}$. Substituting these into the equation: $v + x \frac{dv}{dx} = 1 + v + v^2$. Simplifying,we get $x \frac{dv}{dx} = 1 + v^2$. Separating the variables: $\frac{dv}{1 + v^2} = \frac{dx}{x}$. Integrating both sides: $\int \frac{dv}{1 + v^2} = \int \frac{dx}{x}$. This yields $	an^{-1}(v) = \log |x| + c$. Substituting $v = \frac{y}{x}$ back,we get $	an^{-1} \left( \frac{y}{x} ight) = \log |x| + c$.

The solution of the differential equation $x^2 \frac{dy}{dx} = x^2 + xy + y^2$ is

Similar Questions

The general solution of the differential equation $(3y - 7x + 7)dx + (7y - 3x + 3)dy = 0$ is

The solution of the differential equation $x y^{\prime} = 2 x e^{-y / x} + y$ is

The general solution of the differential equation $(x \sin \frac{y}{x}) dy = (y \sin \frac{y}{x} - x) dx$ is

Find the particular solution satisfying the given condition: $(x+y) dy + (x-y) dx = 0$; $y=1$ when $x=1$.

The solution of the differential equation $\frac{dy}{dx} = \frac{x + y}{x - y}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module