# Inhomogenous differential equation of second order

• Chromosom
In summary, the conversation discusses a method for solving the equation y'' + y = 1/sin x, with one person suggesting the use of variation of constants. The other person confirms that this method works and offers a tip for solving the equation.
Chromosom

## Homework Statement

$$y^{\prime\prime}+y=\frac{1}{\sin x}$$

## The Attempt at a Solution

I solved the homogenous equation: $$y=C_1\sin x+C_2\cos x$$, and then I tried to use method of variable the constant. But the equation system is rather hard. Do you know any other method that can be used here?

The method of variation of constants works well. Show what you have done.

ehild

1 person
Yeah, it does. Thanks for your answer. I started to calculate it and I got it. For future visitors: you must multiplicate first equation by $$\cos x$$ and second by $$\sin x$$ and try to reduce it.

## 1. What is an inhomogeneous differential equation of second order?

An inhomogeneous differential equation of second order is a type of differential equation that involves a second derivative of a function, and also contains a non-zero function on the right-hand side. This non-zero function is referred to as the forcing function or the inhomogeneity of the equation.

## 2. How is an inhomogeneous differential equation of second order different from a homogeneous differential equation of second order?

A homogeneous differential equation of second order only contains the derivatives of a function, without a non-zero function on the right-hand side. This means that the solution to a homogeneous equation will only have one arbitrary constant, while the solution to an inhomogeneous equation will have two arbitrary constants.

## 3. What are the methods for solving inhomogeneous differential equations of second order?

The most common methods for solving inhomogeneous differential equations of second order are the method of undetermined coefficients and the method of variation of parameters. The method of undetermined coefficients involves guessing a particular solution based on the form of the forcing function, while the method of variation of parameters involves finding a solution that satisfies both the differential equation and a complementary equation.

## 4. Can all inhomogeneous differential equations of second order be solved analytically?

No, not all inhomogeneous differential equations of second order can be solved analytically. Some equations may be too complex or do not have a closed-form solution, in which case numerical methods may be used to approximate the solution.

## 5. How are inhomogeneous differential equations of second order used in science?

Inhomogeneous differential equations of second order are used in many areas of science, including physics, engineering, and biology. They can be used to model a wide range of phenomena, such as the motion of objects under the influence of external forces, the behavior of electrical circuits, and the growth of populations. Solving these equations allows scientists to make predictions and understand how systems will behave under different conditions.

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