By multiplying with e^{int P dx} on both side | vedclass

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

Question

(C) The given linear differential equation is $\frac{dy}{dx} + P(x)y = Q(x)$.Multiplying both sides by the integrating factor $I.F. = e^{\int P dx}$,w

Vedclass · Accepted Answer

$e^{\int P dx}$

Vedclass · Answer

(C) The given linear differential equation is $\frac{dy}{dx} + P(x)y = Q(x)$.Multiplying both sides by the integrating factor $I.F. = e^{\int P dx}$,we get:$e^{\int P dx} \frac{dy}{dx} + y P(x) e^{\int P dx} = Q(x) e^{\int P dx}$.We know that the derivative of the product of $y$ and the integrating factor is:$\frac{d}{dx}(y e^{\int P dx}) = e^{\int P dx} \frac{dy}{dx} + y \frac{d}{dx}(e^{\int P dx})$.Since $\frac{d}{dx}(e^{\int P dx}) = e^{\int P dx} P(x)$,the expression becomes:$\frac{d}{dx}(y e^{\int P dx}) = e^{\int P dx} \frac{dy}{dx} + y e^{\int P dx} P(x)$.Comparing this with the given form $\frac{d}{dx}(y f(x))$,we identify that $f(x) = e^{\int P dx}$.

By multiplying with $e^{\int P dx}$ on both sides of the equation $\frac{dy}{dx} + P(x)y = Q(x)$,the left side of the equation takes the form $\frac{d}{dx}(y f(x))$,then $f(x) =$

Similar Questions

The integrating factor of the differential equation $(1+y+x^{2}y)dx + (x+x^{3})dy = 0$ is

An integrating factor of the differential equation $x \frac{dy}{dx} + y \log x = x e^x x^{-\frac{1}{2} \log x}$,$(x > 0)$ is

Find the equation of a curve passing through the origin,given that the slope of the tangent to the curve at any point $(x, y)$ is equal to the sum of the coordinates of the point.

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$ and $\lim_{t \rightarrow x} \left( \frac{t^{2}y(x)-x^{2}y(t)}{x-t} \right) = 3$ for each $x>0$. Then $2y(2)$ is equal to

The general solution of $y \frac{dy}{dx} + by^2 = a \cos x$ for $0 \leq x < 1$ is (where $c$ is an arbitrary constant):

Vedclass Test Series

Exam Paper Generator

Online Exam Module