# Second order differential equation using substitution

• Froskoy
In summary, the given differential equation can be simplified by using the substitution x = cos(\theta) and applying the chain rule. The final expression after differentiation is -sin(\theta)*(dy/dx) + cos^2(\theta)*(d^2y/dx^2). To solve the differential equation, further manipulation and potentially forming an auxiliary equation may be necessary.
Froskoy

## Homework Statement

$$\sin\theta\frac{d^2y}{d\theta^2}-\cos\theta\frac{dy}{d\theta}+2y\sin^3\theta=0$$

## Homework Equations

Use the substitution $x=\cos\theta$

## The Attempt at a Solution

I started off by listing:

$$x=\cos\theta\\ \frac{dx}{d\theta}=-\sin\theta\\ \frac{d^2x}{d\theta^2}=-\cos\theta\\$$

But don't know whether this helps, or where to go next. Could someone please give me a hint at how to approach this, I'd prefer not to have a full solution, but really am desperate for a starting point!

With very many thanks,

Froskoy.

Since you are differentiating with respect to $\theta$, you need to use the chain rule:
$$\frac{dy}{d\theta}= \frac{dy}{dx}\frac{dx}{d\theta}= -sin(\theta)\frac{dy}{dx}$$
Then
$$\frac{d^2y}{d\theta^2}$$
$$= \frac{d}{d\theta}(\frac{dy}{d\theta})$$
$$= \frac{d}{d\theta}(cos(\theta)\frac{dy}{dx}$$
$$= -sin(\theta)\frac{dy}{dx}+ cos(\theta)\frac{d}{d\theta}\frac{dy}{dx}$$
$$= -sin(\theta)\frac{dy}{dx}+ cos^2(\theta)\frac{d^2y}{dx^2}$$

Last edited by a moderator:
Thanks very much! Have got it now!

Hi! I was wondering why
d/dθ (dy/dθ ) does not =d/dθ (-sin(θ) dy/dx)?

Also, once you have the relevant expressions how do you solve the differential equation? Do you have to form the auxilliary equation? I tried to do that but it doesn't work out nicely.

Thanks!

## 1. What is a second order differential equation?

A second order differential equation is a mathematical equation that involves the second derivative of a function. It is commonly used to model physical systems such as motion, heat transfer, and electrical circuits.

## 2. How is substitution used in solving second order differential equations?

Substitution is a technique used to simplify a second order differential equation by replacing the original variable with a new variable. This new variable is chosen in such a way that it reduces the equation to a simpler form, making it easier to solve.

## 3. What is the process for solving a second order differential equation using substitution?

The process for solving a second order differential equation using substitution involves the following steps:

• Identify the original variable and choose a new variable to substitute it with.
• Calculate the first and second derivatives of the new variable with respect to the original variable.
• Substitute the derivatives and the new variable into the original equation.
• Solve the resulting first order differential equation.
• Substitute the solution back into the original equation to find the final solution.

## 4. What are some common applications of second order differential equations?

Second order differential equations are used in various fields of science and engineering to model real-world phenomena. Some common applications include:

• Harmonic motion of a pendulum or spring.
• Heat transfer in objects.
• Electrical circuits and systems.
• Population growth and decay.

## 5. Are there any limitations to using substitution in solving second order differential equations?

While substitution can be a useful technique for simplifying second order differential equations, it may not always be applicable or effective. In some cases, other methods such as variation of parameters or Laplace transforms may be more suitable for solving these equations. Additionally, substitution may not work for equations with variable coefficients or non-linear terms.

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