# Doubt related to formation of a differential equation

• cheapstrike
In summary: So I'll just type it all out again.In summary, the order of the differential equation is equal to the number of arbitrary constants.
cheapstrike

## Homework Statement

Find the order of the differential equation of y=C1sin2x+C2cos2x+C3.

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## The Attempt at a Solution

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I read in my book that the order of the differential equation is equal to the number of arbitrary constants but the answer given is 2.

Btw I have uploaded two pics with two methods I tried.
1st Pic - http://imgur.com/Py7DTgp
2nd Pic - http://imgur.com/5pdcWq1

In first picture, I calculated upto 3rd differential and obtained a differential equation.

In second picture, I differentiated both sides w.r.t. x and then sent the sin2x term, which I was getting in RHS, to LHS and wrote (1/sin2x) as cosec2x. Then I differentiated both sides again w.r.t. x. In this way, both C1 and C2 which remained after calculating 1st derivative become zero.

Which one is correct method? If it's neither, then what's the right method?

cheapstrike said:
I read in my book that the order of the differential equation is equal to the number of arbitrary constants but the answer given is 2.
Your function only has two arbitrary constants. The terms are not linearly independent.

Orodruin said:
Your function only has two arbitrary constants. The terms are not linearly independent.
Can you elaborate please? I mean how is C3 not an arbitrary constant?

cheapstrike said:
Can you elaborate please? I mean how is C3 not an arbitrary constant?
The functions that they multiply are linearly dependent. You can rewrite ##\sin^2 x## in terms of cos(2x) and a constant.

cheapstrike
Orodruin said:
The functions that they multiply are linearly dependent. You can rewrite ##\sin^2 x## in terms of cos(2x) and a constant.
Thanks

cheapstrike said:

## Homework Statement

Find the order of the differential equation of y=C1sin2x+C2cos2x+C3.

Can you see how to re-write ##C_1 \sin^2(x) + C_2 \cos 2x + C_3## as ##A \cos2x + B##? Working with the latter form is easier.

Note added in edit: I see that once again PF has fooled me, by only displaying post #4 after I pressed submitted my answer.

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives, which represent the rate of change of a function, to express a relationship between a function and its independent variables.

## 2. How is a differential equation formed?

Differential equations are formed by expressing a relationship between a function and its derivatives. This relationship can be expressed in terms of the dependent variable, the independent variable, and the derivatives of the dependent variable with respect to the independent variable.

## 3. What is the purpose of using differential equations?

Differential equations are used to model and solve a wide range of problems in various fields such as physics, engineering, economics, and biology. They provide a mathematical framework for understanding and predicting the behavior of complex systems.

## 4. What are the types of differential equations?

There are several types of differential equations, including ordinary differential equations, which involve only one independent variable, and partial differential equations, which involve multiple independent variables. Other types include linear and nonlinear differential equations, as well as first-order and second-order differential equations.

## 5. How are differential equations solved?

Differential equations can be solved analytically, using mathematical methods, or numerically, using computational techniques. Analytical methods involve finding an explicit solution to the equation, while numerical methods use algorithms to approximate the solution. The method used depends on the complexity of the equation and the availability of analytical techniques.

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