ODE- not sure if this is correct

[newtomath]

The problem is :

(y*e^(2xy) +x) + [ b* x*e^(2xy) ] y' =0. Find b so the equation is exact and solve.

I found b=1 and worked the problem to (1/2)e^(2xy) + (x^2/2)e^(2xy) + h(y); where I found h(y) to be simply c.

The answer in the text states that e^(2xy) + x^2 =c

Would I be incorrect to multiply [ (1/2)e^(2xy) + (x^2/2)e^(2xy) = -c ] by 2 to arrive at the answer since c is an arbitrary constant?

Thanks

 

[{???}]

The problem is :

(y*e^(2xy) +x) + [ b* x*e^(2xy) ] y' =0. Find b so the equation is exact and solve.

I found b=1 and worked the problem to (1/2)e^(2xy) + (x^2/2)e^(2xy) + h(y); where I found h(y) to be simply c.

The answer in the text states that e^(2xy) + x^2 =c

Would I be incorrect to multiply [ (1/2)e^(2xy) + (x^2/2)e^(2xy) = -c ] by 2 to arrive at the answer since c is an arbitrary constant?

Thanks

I believe you are correct in multiplying through that way, absorbing the factor of 2 into the constant. However, it still will not give you the answer the text gives because you will now have e^(2xy)+x^2 e^(2xy)=c, where as the text gives e^(2xy)+x^2=c.

 

[newtomath]

Thanks. My mistake, the second term is (x^2/2) , not (x^2/2)e^(2xy)