# Inroductory differential equations

• w3390
In summary, the student attempted to solve for y's constant solutions by factoring the differential equation, but it always led to integrating a factor with y's with respect to t. They were then asked to do what Ivy said and it led to a solution.
w3390

## Homework Statement

A function y(t) satisfies the differential equation: dy/dt= y^4-7y^3+6y^2. What are the constant solutions to the equation?

## Homework Equations

dy/dx= g(x)*f(y) --> INT([1/f(y)]dx/dy)=INT(g(x))

## The Attempt at a Solution

My first attempt was to factor out a y^2 term and divide it to the other side and take the integral. After a couple more steps, I realized this was not correct. My second attempt was to factor the differential equation into two factors and divide one of the factors to the other side, but this always leaves me integrating a factor with y's with respect to t. I cannot figure out how to attack this problem.

If you want the general solution why don't you just multiply both sides by dt and then divide by y^4-7y^3+6y^2.

That would give you

dy/ y^4-7y^3+6y^2 = dt

So integrate the left side with respect to y, and the right side with respect to t.

w3390 said:

## Homework Statement

A function y(t) satisfies the differential equation: dy/dt= y^4-7y^3+6y^2. What are the constant solutions to the equation?

## Homework Equations

dy/dx= g(x)*f(y) --> INT([1/f(y)]dx/dy)=INT(g(x))

## The Attempt at a Solution

My first attempt was to factor out a y^2 term and divide it to the other side and take the integral. After a couple more steps, I realized this was not correct. My second attempt was to factor the differential equation into two factors and divide one of the factors to the other side, but this always leaves me integrating a factor with y's with respect to t. I cannot figure out how to attack this problem.
You are NOT asked to find a general solution! You are only asked to find the constant solutions. if y is constant then dy/dx= 0.

Can you solve y^4-7y^3+6y^2= 0?

oh wow good catch Ivy! Maybe i need to get some glasses... hah...

Do what Ivy said!

Time to stop doing it?

## 1. What is the purpose of studying introductory differential equations?

Studying introductory differential equations is important because it helps us model and understand real-world phenomena that change over time. These equations are used in various fields such as physics, engineering, economics, and biology to describe the behavior of systems.

## 2. What are differential equations?

Differential equations are mathematical equations that involve the rate of change of a quantity with respect to another quantity. They can be used to describe how a system changes over time, and they often involve functions and their derivatives.

## 3. What are some common applications of introductory differential equations?

Introductory differential equations are used to model and solve problems in a variety of fields. Some common applications include population growth, chemical reactions, heat transfer, and motion of objects under the influence of forces.

## 4. What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations involve functions of one independent variable, while partial differential equations involve functions of multiple independent variables. Stochastic differential equations involve randomness and are used in modeling systems with uncertain behavior.

## 5. What are some techniques for solving introductory differential equations?

There are several techniques for solving differential equations, including separation of variables, substitution, and using integrating factors. Other methods include power series solutions, Laplace transforms, and numerical methods such as Euler's method and Runge-Kutta methods.

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