# Solving a 2nd Order ODE: P^2 - 4xP +6y = 0

In summary, the problem involves finding a common approach for solving an equation by letting P = dy/dx. The equation is then differentiated to get 6P = 4(P+x-1). By dividing by dP/dx, the equation can be simplified to P+x=1+P, leading to the solution of P=1+x and dP/dx=1.

## Homework Statement

$$\left(\frac{dy}{dx}\right)^2 - 4x\frac{dy}{dx} + 6y = 0$$

## Homework Equations

A common approach we have used for similar problems has been to let P = dy/dx

## The Attempt at a Solution

Doing so we have:

$$P^2 - 4xP +6y = 0$$

$$\Rightarrow 6y = 4P(x - P)$$

Differentiating gives:

$$6P = 4\left[P(1 - \frac{dP}{dx}) +\frac{dP}{dx}(x - P)\right] = 0$$Now usually we try to factor this and solve each factor as a linear 1st order EQ in P. However, I am having trouble seeing a nice way to factor this, that makes each factor linear. All I can get to is

$$-2\left[P+2\frac{dP}{dx} - 2x\frac{dP}{dx} + 2P\frac{dP}{dx}\right] = 0$$Any thoughts on what to do with the bracketed term to get 2 linear EQs out of the deal?

Thanks

A:$$6P = 4\left[P(1 - \frac{dP}{dx}) +\frac{dP}{dx}(x - P)\right] = 0$$$$\Rightarrow 6P=4\left[P-\frac{dP}{dx}+x\frac{dP}{dx}-P\frac{dP}{dx}\right]$$$$\Rightarrow 6P=4\left[P+x\frac{dP}{dx}-\left(1+P\right)\frac{dP}{dx}\right]$$$$\Rightarrow \frac{6P}{4}=P+x\frac{dP}{dx}-\left(1+P\right)\frac{dP}{dx}$$$$\Rightarrow P+x\frac{dP}{dx}-\left(1+P\right)\frac{dP}{dx}=0$$Divide by \$\frac{dP}{dx}$$\Rightarrow P+x-\left(1+P\right)=0$$$$\Rightarrow P+x=1+P$$$$\Rightarrow P=1+x$$$$\Rightarrow \frac{dP}{dx}=1$$

## 1. What is a 2nd order ODE?

A 2nd order ODE (ordinary differential equation) is a type of mathematical equation that involves the derivatives of a function with respect to a single independent variable. In other words, it is an equation that relates a function with its derivatives.

## 2. How do you solve a 2nd order ODE?

To solve a 2nd order ODE, you need to follow a systematic process. First, you need to identify the type of equation (homogeneous or non-homogeneous) and then use appropriate methods (such as substitution or variation of parameters) to find the general solution. Finally, you can use initial conditions to find the particular solution.

## 3. What is the role of P in a 2nd order ODE?

In a 2nd order ODE, P represents the first derivative of the function. It is also known as the independent variable or the parameter. P is used to differentiate the function and its derivatives in the equation.

## 4. Why is it important to solve a 2nd order ODE?

Solving a 2nd order ODE is important in many fields of science and engineering. It allows us to model and understand complex systems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits. It also helps us make predictions and solve real-world problems.

## 5. What are some applications of 2nd order ODEs?

2nd order ODEs have various applications in fields such as physics, engineering, chemistry, and biology. They are used to describe the motion of objects under the influence of forces, the flow of fluids in pipes and channels, the behavior of electrical circuits, and the growth of populations. They are also used in mathematical models for predicting natural phenomena, such as weather patterns and population dynamics.

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