# Solving multivariable equation with integration

• gingermom
In summary, the conversation is about solving the differential equation dy/dx = (1+x)/xy with the initial condition y(1) = -4. The individual asking the question initially struggled with isolating x and using the inverse, but eventually found the correct approach of solving for the constant C and subbing it back into the solution.
gingermom
Gold Member

## Homework Statement

dy/dx = (1+x)/xy solve y(1) = -4

## The Attempt at a Solution

What threw me was the solve for if y(1) = -4

I grouped variables and then integrated both sides and solved for y.

(1/2)y^2 = ln|x|+x+c
y=+- √2ln|x|+x+c

I then switched the x and y to get the inverse of the equation and set that equal to -4

-4=-+ √2ln1+1+c
16= 2*(1+c)
8-1=c
7=c

However, I am not sure this was the correct approach. I could not figure out how to isolate x in the beginning, but I am not 100 percent sure of my logic of using the inverse, and whether I applied it appropriately. If this is way off base, can you point me in the correct direction? We have not had any problems quite like this and I am not sure I am making the correct leap in combining concepts.

Thanks.

gingermom said:

## Homework Statement

dy/dx = (1+x)/xy solve y(1) = -4

## The Attempt at a Solution

What threw me was the solve for if y(1) = -4

I grouped variables and then integrated both sides and solved for y.

(1/2)y^2 = ln|x|+x+c
y=+- √2ln|x|+x+c

I then switched the x and y to get the inverse of the equation and set that equal to -4

-4=-+ √2ln1+1+c
16= 2*(1+c)
8-1=c
7=c

However, I am not sure this was the correct approach. I could not figure out how to isolate x in the beginning, but I am not 100 percent sure of my logic of using the inverse, and whether I applied it appropriately. If this is way off base, can you point me in the correct direction? We have not had any problems quite like this and I am not sure I am making the correct leap in combining concepts.

Thanks.

##y= +- \sqrt{2ln|x| + 2x + 2c}##

Then ##y(1) = -4## would indeed imply ##c = 7##.

Then simply subbing ##c## into your solution from prior would be the last thing to do.

Thanks - Yes, sorry about that I had used parenthesis but they disappeared and I didn't notice. So using the inverse to find C was the right thing to do. Yeah, maybe I am getting this after all! I was so focused on finding C I forgot about putting it back in. Should have posted that part, too I guess. First time here. Will try and do better if ( I probably should say when) I need additional help. Thanks again.

## 1. What is a multivariable equation?

A multivariable equation is an equation that contains more than one variable. This means that there are multiple unknown quantities that need to be solved for.

## 2. How is integration used to solve multivariable equations?

Integration is a mathematical process that involves finding the area under a curve. In the context of multivariable equations, integration is used to find the values of the unknown variables that satisfy the equation.

## 3. What is the difference between single-variable and multivariable equations?

Single-variable equations have only one unknown quantity, while multivariable equations have multiple unknown quantities. This means that solving a multivariable equation requires finding values for all the variables, whereas a single-variable equation only requires finding one value.

## 4. Can all multivariable equations be solved using integration?

No, not all multivariable equations can be solved using integration. Some equations may require other methods such as substitution or elimination.

## 5. Are there any limitations to solving multivariable equations using integration?

Yes, there are limitations to solving multivariable equations using integration. For example, the variables in the equation must be continuous functions, and the boundaries of integration must be known and finite.

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