Integrating differential equations that have ln

In summary, the conversation revolves around the rewriting of a differential equation solution and the confusion surrounding plugging in coordinates to find a particular solution. The general solution represents all possible functions, but choosing the correct branch (positive or negative values for C) is necessary to obtain the correct portion of the graph. Plugging in coordinates to the general solution may result in missing a portion of the graph, and the use of absolute values is important when integrating a ln derivative. Ultimately, choosing one branch is necessary to ensure that y is a function with a unique value for each value of x.
  • #1
ecoo
86
2
Hey guys, I have a question concerning the rewriting of a differential equation solution.

seperation of variables.png


In the example above, they rewrite [y=(plus/minus)e^c*sqrt(x^2+4)] as [y=C*sqrt(x^2+4)]. I understand that the general solution we get as a result represents all the possible functions, but if we were to attempt to find a particular solution given an initial condition (a point on the graph of the equation), I would think that we would plug in the coordinate into not the general solution but the one above it with the (plus/minus). The problem is that in some of the problems that is not the case - you plug in the coordinate into the general solution to find your equation.

The reason I would plug in the coordinate into the (plus/minus) equation is because that is the original equation contains above the x-axis values and below the x-axis values (when you graph the resulting equation). But if you plug a coordinate into the general solution, you only get the top portion of bottom portion of the graph depending on if y were less than or greater than 0 (because of the absolute value symbol). So plugging into the general solution kind of makes you lose a portion of the equation's graph.

Hopefully you guys can help clear up this confusion for me. If you need any clarification, please ask.

Thanks!
 
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  • #2
One obtains the lower portion of the graph from the general solution (last line of your post) by choosing C<0.
 
  • #3
andrewkirk said:
One obtains the lower portion of the graph from the general solution (last line of your post) by choosing C<0.

Does that mean we would be missing the bottom portion of the graph if we only choose C > 0. To see the whole equation, we'd have to choose C and -C. If we only chose C, shouldn't we mention that we are only seeing the y > 0 portion of the graph?

I guess my confusion comes from problems like this, where they don't put the y in absolute value (I don't know why), which you are supposed to do when integrating a ln derivative. By not putting the y in derivatives, aren't they losing the bottom portion of the graph? And if they did put the y in absolute values, when solving for the particular solution given a coordinate, wouldn't you have to use the (plus/minus) equation to get the top and bottom?
problem answer 1.png
 
  • #4
You have to choose one branch or the other, or else y is not a function. A function of x must have a unique value for each value of x.
 
  • #5
andrewkirk said:
You have to choose one branch or the other, or else y is not a function. A function of x must have a unique value for each value of x.

Thank you! Succinct and insightful :)
 

Related to Integrating differential equations that have ln

What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It can be used to model a wide range of phenomena in various fields such as physics, engineering, and economics.

Why do we need to integrate differential equations?

Integrating differential equations allows us to find the original function when only its derivatives are known. This is useful for solving problems in physics, engineering, and other fields where the relationship between variables is described by a differential equation.

What is ln in a differential equation?

ln is the natural logarithm function, which is used to solve differential equations that involve exponential growth or decay. It is the inverse of the exponential function and is commonly used in calculus and other areas of mathematics.

What are the steps for integrating a differential equation with ln?

The steps for integrating a differential equation with ln include first isolating the variable with the derivative, then taking the natural logarithm of both sides. From there, you can use algebraic manipulation and basic integration techniques to solve for the original function.

What are some real-life applications of integrating differential equations with ln?

Differential equations with ln are commonly used to model real-life phenomena such as population growth, radioactive decay, and chemical reactions. They are also used in engineering to analyze systems that involve exponential growth or decay, such as the charging and discharging of capacitors in electrical circuits.

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