# Differential equations question - possible integrating factor?

• tourjete
In summary, the problem is to solve the differential equation 3e5xdy/dx = -25x/y2 with the initial condition y(0) = 1. The equation is separable, and the first approach of putting everything with an x or dx on one side and a y or dy on the other side is correct. However, the constant value of 0 obtained from this method is incorrect according to the online homework system. The use of an integrating factor is not applicable in this case due to the presence of y2 instead of y. Additional assistance may be needed to find the correct solution.
tourjete

## Homework Statement

Solve the differential equation with the initial condition y(0) = 1

## Homework Equations

3e5xdy/dx = -25x/y2

## The Attempt at a Solution

First I tried putting everything with an x or dx on one side and a y or dy on the other side, and solved for C. I got 0 as the constant, but when I plugged it into the online homework system it said I was wrong.

I thought about using an integrating factor, but that can only be used when the equation is in the form dy/dt + a(t)y = r(t), right? Meaning I can't use it here because there's a y2 instead of a y?

The equation is separable, so your first approach is the right one. Can you show some of your work?

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It describes how a quantity changes over time or space and is an essential tool in many scientific fields, including physics, engineering, and economics.

## 2. What is an integrating factor in differential equations?

An integrating factor is a function that is multiplied to both sides of a differential equation to make it easier to solve. It is used to convert a non-exact differential equation into an exact one, which can then be solved using traditional mathematical techniques.

## 3. How do you find an integrating factor?

To find an integrating factor, the differential equation must first be written in standard form, which is dy/dx + P(x)y = Q(x). Then, the integrating factor is calculated by taking the exponential of the integral of P(x)dx. The integrating factor is then used to multiply both sides of the equation, making it easier to solve.

## 4. Why do we need integrating factors?

Integrating factors are needed because not all differential equations can be solved using traditional mathematical techniques. By multiplying an integrating factor, the differential equation can be transformed into a simpler form that can be solved using methods such as separation of variables or integration by parts.

## 5. Can all differential equations be solved using integrating factors?

No, not all differential equations can be solved using integrating factors. Some equations may require more advanced mathematical techniques or may not have a solution at all. In general, integrating factors are most useful for linear first-order differential equations, but may also be used for some higher-order equations.

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