Is The solution of differential equation correct?

Test Preparation AP Mathematics & Computer Science
1

<p>Q: Solve the differential equation :
(x+y+1)(dy/dx) = 1</p>

<p>Ans : I put (x+y)=t
=> y=t-x</p>

<p>(t+1)(d(t-x)/dx) = 1</p>

<p>(t+1)(dt/dx - 1) = 1</p>

<p>(t+1)(dt/dx) = t+2</p>

<p>[(t+1)/(t+2)]*(dt/dx) = dx</p>

<p>integrating both sides,</p>

<p>int[ t/(t+2) + 1/(t+2) ]dt = int.dx</p>

<p>now the second term in the LHS can be integrated directly, and so can the RHS.</p>

<p>but the term t/(t+2) has to be integrated separately by adding and subtracting 2 in numerator.</p>

<p>=> (t+2-2)/(t+2)</p>

<p>=> 1 - 2/(t+2)</p>

<p>and so this can now be solved too. </p>

<p>=> int.[ 1-2/(t+2) + 1/(t+2) ]dt = int.dx</p>

<p>=> int.[ 1 - 1/(t+2) ]dt = int.dx</p>

<p>=> t - log(t+2) = x +C</p>

<p>substituting t= x+y</p>

<p>x+y - log(x+y+2) = x +C</p>

<p>y= log(x+y+2) +C</p>

<p>which i think is the answer.
am I correct? I dont have the answers so if some1 can verify, It would be a great.</p>

2

<p>Looks alright, but the closed form solution for y can’t be expressed in terms of elementary functions.</p>

<p>A slightly shorter way to integrate the LHS is to see that (t+1)/(t+2) = (t+2)/(t+2) - 1/(t+2).</p>

3

<p>Thanks Man! :)</p>

<p>and i didn’t see that shortcut, nice thinking.</p>