# Difficult first order linear differential equation

• MHB
• WMDhamnekar
In summary, the conversation is about solving the differential equation $y'=\dfrac{x^3-y^3}{x-y}$ using Wolfram Development Platform (essentially Mathematica). However, the resulting answer is complicated and involves Hermite $H_n(x)$ functions and the hypergeometric $_1F_1$ function. The speaker asks for context and what has been tried so far. The respondent then provides a simplified version of the solution using a differential equation solver, reducing the expression to $y'=x^2+x y+y^2$.
WMDhamnekar
MHB
Hello, I want to solve the following differential equation. $y'=\dfrac{x^3-y^3}{x-y}$. How to solve it?

Wolfram Development Platform (essentially Mathematica) gives a horrendous answer: extremely complicated with Hermite $H_n(x)$ functions and the hypergeometric $_1F_1$ function. In what context did this problem come up, and what have you tried?

Ackbach said:
Wolfram Development Platform (essentially Mathematica) gives a horrendous answer: extremely complicated with Hermite $H_n(x)$ functions and the hypergeometric $_1F_1$ function. In what context did this problem come up, and what have you tried?

I got the following answer from my differential equation solver.

$y(x)=\frac{x^3}{6}(C_1(C_1+1)+3C_1+2)+\frac{x^5}{120}(C^3_1+C_1(74C_1+9)+31C_1+8)+C_1+C_1x+\frac{3C_1x^2}{2}+\frac{C_1x^4}{12}(C_1+4)+\mathcal{O}$

Last edited by a moderator:
You can reduce the expression to this

y'=x^2 + x y + y^2

## What is a first order linear differential equation?

A first order linear differential equation is a type of differential equation where the highest derivative of the dependent variable is raised to the first power and the equation can be written in the form y' + P(x)y = Q(x), where P(x) and Q(x) are functions of x.

## Why is a first order linear differential equation difficult?

A first order linear differential equation can be difficult because it involves finding an unknown function that satisfies the equation and can require advanced mathematical techniques to solve.

## What is the general solution to a first order linear differential equation?

The general solution to a first order linear differential equation is a function that includes all possible solutions to the equation. It is typically expressed as y = Ce^(-∫P(x)dx) + ∫Q(x)e^(∫P(x)dx)dx, where C is a constant of integration.

## How is a first order linear differential equation solved using the method of integrating factors?

The method of integrating factors involves multiplying both sides of the differential equation by an integrating factor, typically a function of x, in order to make the equation easier to solve. This method is particularly useful for first order linear differential equations with non-constant coefficients.

## What are some real-world applications of first order linear differential equations?

First order linear differential equations have many real-world applications, such as modeling population growth, predicting the spread of diseases, and analyzing the flow of fluids. They are also used in engineering and physics to understand systems that involve changing rates over time.

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