Question:
The general solution of the differential equation $\frac{dy}{dx}+\frac{2}{x}y=x^{2}$ is
The general solution of the differential equation $\frac{dy}{dx}+\frac{2}{x}y=x^{2}$ is
Updated On: Aug 15, 2022
- $y=cx^{-3}-\frac{x^{2}}{4}$
- $y=cx^{3} - \frac{x^2}{4}$
- $y=cx^{2} + \frac{x^3}{5}$
- $y=cx^{-2} + \frac{x^3}{5}$
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The Correct Option is D
Solution and Explanation
Given differential equation is $\frac{dy}{dx}+\frac{2}{x}.y=x^{2}$
This is of the linear form.
$\therefore P=\frac{2}{x}, Q=x^{2}$
$I.F.=e^{\int \frac{2}{x}dx}=e^{log\,x^2}=x^{2}$
Solution is
$yx^{2}=\int\,x^{2}\,.x^{2}\,dx+c=\frac{x^{5}}{5}+c$
$y=\frac{x^{3}}{5}+cx^{-2}$
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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