# Solve Differential Eq.: dy/dx= 1/(xy+x^2y^3)

• estalniath
In summary, the conversation discusses a differential equation and the various methods used to solve it. The equation is not of a familiar form and appeared in a past examination question. The conversation also mentions the use of substitution and the use of Latex code for mathematical equations.
estalniath
Hello guys,

Which method can we use to solve the differential equation below?
dy/dx= 1/(xy+x^2y^3)

It doesn't seem to be of any form which I had studied before(linear differential equations,bernoulli,exact differential,homogeneous,seperable equations) for first order differential equations yet it appeared in one of out past examination questions

It's easy.For me

$$\frac{dy}{dx}=\frac{1}{xy+x^{2}y^{3}}$$

Therefore

$$\frac{dx}{dy}=yx+y^{3}x^{2}$$

Make the substitution

$$x=\frac{1}{u}$$

,under which

$$\frac{dx}{dy}=-\frac{1}{u^{2}}\frac{du}{dy}$$

Therefore the new ODE is

$$-\frac{1}{u^{2}}\frac{du}{dy}=\frac{y}{u}+y^{3}\frac{1}{u^{2}}$$

equivalently

$$\frac{du}{dy}=-yu-y^{3}$$

I trust you can take it from here.It's an nonhomegenous linear ODE...(the homogenous eq is separable).

Daniel.

Wow! That was a fast reply. =) Thank you for replying and how do you type the mathematical equations in between? I don't seem to see any functions in this thread which allows us to type mathematical functions?

Write formulas in latex code and use preview option for checking it b4 clicking submit.

Daniel.

I see. Thank you for the advice =)

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to model and solve a wide range of physical, biological, and economic phenomena.

## 2. How do you solve a differential equation?

To solve a differential equation, you need to find a function that satisfies the equation. This can be done through various methods such as separation of variables, integrating factors, and using specific formulas for certain types of differential equations.

## 3. What is the order of a differential equation?

The order of a differential equation is the highest order of derivative present in the equation. For example, a first-order differential equation has only first derivatives, while a second-order differential equation has second derivatives.

## 4. Can you explain the given differential equation?

The given differential equation is a first-order separable differential equation. This means that it can be rewritten as the product of two functions, one involving only x and the other involving only y, and then integrated to find the general solution.

## 5. What is the general solution to the given differential equation?

The general solution to the given differential equation is y = C/(x+xy^3), where C is a constant of integration. However, if initial conditions are given, the constant C can be determined and a particular solution can be found.

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