Question:
The solution of the differential equation log x $\frac{d y}{d x}+\frac{y}{x} =$ sin 2x is
The solution of the differential equation log x $\frac{d y}{d x}+\frac{y}{x} =$ sin 2x is
Updated On: Apr 19, 2024
- $y log \left|x\right| =C-\frac{1}{2} cos\, x$
- $y log \left|x\right| =C+\frac{1}{2} cos\, 2 x$
- $y log \left|x\right| =C-\frac{1}{2} cos\, 2 x$
- $xy log \left|x\right| =C-\frac{1}{2} cos\, 2 x$
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The Correct Option is C
Solution and Explanation
$\frac{d y}{d x}+\frac{y}{x \, log\, x }=\frac{sin \, 2x}{log\, x}$
$I.F. =e^{\int \frac{d x}{x \, log \, x}}$
$\therefore\quad I.F. = e^{\int \frac{1}{t} dt}=e^{log\, t} =t=log \left|x\right|$
solution is given by
$y \left(I.F.\right)=\int \left(I.F.\right) dx+C$
$y log \left|x\right|= \int \frac{sin\, 2x}{log\, \left|x\right|} \left(log \left|x\right|\right) dx+C$
$=-\frac{cos\, 2x}{2}+C$
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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