Question:
The integrating factor of the differential equation $ \left(1+x^{2}\right)\frac{dy}{dx}+xy=cos\,x $ is equal to
The integrating factor of the differential equation $ \left(1+x^{2}\right)\frac{dy}{dx}+xy=cos\,x $ is equal to
Updated On: Apr 29, 2024
- $ \sqrt{1+x} $
- $ \sqrt{1+2x^{2}} $
- $ \sqrt{1+x^{2}} $
- $ \sqrt{2+x^{2}} $
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The Correct Option is C
Solution and Explanation
We have, $(1 + x^2) \frac{dy}{dx} + xy = cos\,x$
$\Rightarrow \frac{dy}{dx} + (\frac{x}{1+x^2}) y =\frac{cos\,x}{1+x^2}$
$\therefore I.F. = e^{\left(\frac{1}{2}\int\frac{2x}{1+x^{2}}dx\right)} = e^{\frac{1}{2}log\left(1+x^{2}\right)} $
$= \sqrt{1+x^{2}}$
$\Rightarrow \frac{dy}{dx} + (\frac{x}{1+x^2}) y =\frac{cos\,x}{1+x^2}$
$\therefore I.F. = e^{\left(\frac{1}{2}\int\frac{2x}{1+x^{2}}dx\right)} = e^{\frac{1}{2}log\left(1+x^{2}\right)} $
$= \sqrt{1+x^{2}}$
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations