Differential Equation using Integrating Factor

[CFDFEAGURU]

Homework Statement

The ODE is

(1+x)*dy/dx - xy = x+x^2

Homework Equations

The method of solution is to be through the use of the integration factor.

The Attempt at a Solution

First, I divided each side by (1+x) to produce

dy/dx - xy/(1+x) = x

then factor out the x on the LHS to produce

dy/dx - x * (y/(1+x)) = x

then divide both sides by x to produce

dy/dx - y/(1+x) = 0

now move the -y/(1+x) to the RHS to produce

dy/dx = y/(1+x)

That is all the farther I have progressed. Am I correct so far?

Thanks
Matt

 

[HallsofIvy]

Yes, you are correct so far by what progress is that? How does that help you find the integrating factor?

The left side of your equation is (1+x)(dy/dx)- xy. The whole point of an "integrating factor" is that it is a function u(x) such that u(x)(1+x)(dy/dx)- uxy= d(u(x)(1+x)y)/dx.
Go ahead and use the product rule on the right side. That gives u(1+x)(dy/dx)+ u(1)y+ (du/dx)(1+x)y and that must be equal to u(x)(1+x)(dy/dx)- uxy. Well, the "u(1+x)(dy/dx)" cancels immediately leaving u(x)y+ (1+x)y(du/dx)= -uxy so (1+x)y(du/dx)= -xy(1+ u). Now the "y"s cancel so (1+x)(du/dx)= -x(1+u) which is a separable equation. du/(1+u)= -dx/(x(1+x)). Integrate that to find the integrating factor.

 

[CFDFEAGURU]

By progress, I was meaning the reformatting of the original equation. I believed that this equation is seperable but the assignment was to use the integrating factor method.

I would rather take the path of the seperable equation.

Starting with

dy/dx = y/(1+x)

multiplying both sides by dx produces

dy = y/(1+x)dx

dividing both sides by y produces

dy/y = dx/(1+x)

integrating both sides produces

ln(y) = ln(1+x) + c

Is this correct so far?


Now proceeding forward with your instructions to integrate

du/(1+u) = -dx/(x(1+x)

upon integration of both sides

ln(1+u) = -ln(x)+ln(1+x)

now the integrating factor would be

e^(ln(1+u))

upon integration would be

1+u

Am I correct so far?

Thanks
Matt

 

[CFDFEAGURU]

OK. I think I have the solution.

With the integrating factor of 1+x and using the equation

dy/dx -y/(1+x) = 0

multiplying both sides by the integrating factor produces

(1+x)*dy/dx - (1+x)*y/(1+x) = 0

which is equal to

(1+x) * dy/dx - y = 0

upon integrating

(1+x)*y = c

solving for y produces

y = c/(1+x)

Am I correct?

Thanks
Matt