Solving Differential Equations: Understanding the Steps

In summary, the conversation is about someone looking for help solving two differential equations: x'=-x and x'=x^2, x(0)=x0. The solutions to these equations are x(t)=e^-tx0 and x(t)=x0/(1-x0t). The speaker is asking for clarification on the steps used to arrive at these solutions and suggests providing more information or referencing a textbook. They also mention that there are various approaches to solving these equations and knowing the desired approach may be helpful.
  • #1
kingpen123
1
0
I am looking for help solving these two differential equations:

1. x'=-x

2. x'=x2, x(0)=x0

The solutions are x(t)=e-tx0, and x(t)=x0/(1-x0t).

I just don't understand what steps were being done to get those solutions. If someone could point me in the right starting point or show me some steps to get these solutions it would be much appreciated.
 
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  • #2
Hi Kingpen,

Can you maybe show us some more information or for example point us to your textbook? The problem is that there are many ways to "solve" these equations. In particular, you can "guess" the solution and then show that it works and you can even use the differential equation x' = x with initial value x(0) = 1 as the definition of the function f(t) = et. So knowing by which approach / on which level you would like to solve these equations may help.
 

FAQ: Solving Differential Equations: Understanding the Steps

What is a differential equation?

A differential equation is a mathematical equation that describes how a function's value changes in relation to its input variables. It involves the use of derivatives, which represent the rate of change of a function at a given point.

Why are differential equations important?

Differential equations are important because they are used to model many real-world phenomena, such as population growth, heat flow, and electric circuits. They allow scientists to make predictions and solve problems in fields such as physics, engineering, and economics.

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 one independent variable, while partial differential equations involve multiple independent variables. Stochastic differential equations involve random variables.

How are differential equations solved?

Differential equations can be solved analytically or numerically. Analytical solutions involve finding an explicit formula for the solution, while numerical solutions involve using algorithms to approximate the solution. Different methods, such as separation of variables, variation of parameters, and Euler's method, can be used to solve differential equations.

What are some applications of differential equations?

Differential equations have many applications in science and engineering. They are used to model physical systems, such as the motion of objects, chemical reactions, and electrical circuits. They are also used in economics, biology, and medicine to understand and predict various phenomena.

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