Question:
The solution of differential equation
$\frac{dy}{dx}-3y=sin\,2x$ is
The solution of differential equation
$\frac{dy}{dx}-3y=sin\,2x$ is
Updated On: Jul 7, 2022
- $y=e^{-3x}\left[\frac{cos\,2x+3\,sin\,2x}{13}\right]+c$
- $y=e^{-3x}\left(\frac{cos\,2x-3\,sin\,2x}{13}\right)+c$
- $ye^{-3x}=-e^{-3x} \frac{\left(2\,cos\,2x+3\,sin\,2x\right)}{13}+c$
- none of these
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The Correct Option is C
Solution and Explanation
$\frac{dy}{dx}-3y=sin\,2x$
$\Rightarrow$ It is linear equation with $I.F.=e^{\int-3dx}=e^{-3x}$
Required solution is, $y\cdot e^{-3x}=\int e^{-3x}\,sin\,2x\,dx$
$=e^{-3x} \frac{\left(-3\,sin\,2x-2\,cos\,2x\right)}{2^{2}+3^{2}}+c$
$\Rightarrow y\,e^{-3x}=-e^{-3x} \frac{\left(3\,sin\,2x+2\,cos\,2x\right)}{13}+c$
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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