Thread: Solve y'(x)=(4x+3y-1)^2

Hello,
I have solved this differential equation, but I want to know whether all the steps are correct.
y'(x)=(4x+3y-1)^2
Let u=4x+3y-1 then du/dx =4+3y'(x)
This implies y'(x)=(1/3)u'(x)-4/3
Now
(1/3)u'(x)-4/3=u^2
Hence,u'(x)=3u^2+4
du/(3u^2+4)=dx
Integrating both the sides,we get
1/2sqrt(3) arctan(sqrt(3)u)/2=x+c
So my ans is sqrt(3)*(4x+3y-1)= 2tan(2sqrt(3)*x)+c
If I am wrong, reply where I am wrong.

You have made some slight errors. I have corrected them.

Originally Posted by Vinod

Hello,
I have solved this differential equation, but I want to know whether all the steps are correct.
y'(x)=(4x+3y-1)^2
Let u=4x+3y-1 then du/dx =4+3y'(x)
This implies y'(x)=(1/3)u'(x)-4/3
Now
(1/3)u'(x)-4/3=u^2
Hence,u'(x)=3u^2+4
du/(3u^2+4)=dx
Integrating both the sides,we get
1/2sqrt(3) arctan((sqrt(3)u)/2)+k=x+c
So my ans is sqrt(3)*(4x+3y-1)= 2tan((2sqrt(3)*x)+c-k)
If I am wrong, reply where I am wrong.

You don't have to write the constant of integration on both sides and then subtract. That will simply be a single constant.
Vinod, your solution is correct.