# Thread: First Order Homogenous Differential Equation

1. ## First Order Homogenous Differential Equation

By using the substitution y=vx, show that the general solution of the first order homogeneous differential equation (x+y)[dy/dx]= y-x in the case where x>0 is given by y=k, where k is a constant.

Many thanks in advance! 2. Originally Posted by cyt91 By using the substitution y=vx, show that the general solution of the first order homogeneous differential equation (x+y)[dy/dx]= y-x in the case where x>0 is given by y=k, where k is a constant.

Many thanks in advance! Okay, have you tried this at all? You can write the given equation as (x+y)dy= (y- x)dx. If y= vx, then dy= v dx+ x dv. Replace dy with that and replace y with xv and see what you get!

Far better to do it yourself than have someone do it for you.

3. Yes, I did. I came to

arc tan [y/x] + ln {[(y^2 + x^2)^0.5]/[x^2]} = ln {A/x}

where A is a constant.

Now, how do you obtain the expression y=k where k is a constant?

Is my solution correct?

Thanks a lot! 4. Check your question because y = K is not the general solution

If y = K were the solution

dy/dx = 0 = (K - x)/(K+x) is only true for the single value x= K

you're solution matches what i got

5. I've checked the question with my instructor. Yea, the question has problems. It's the book's publisher's mistake. Thanks anyway. Good thing I got the general solution right.

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