# Thread: how to expres in terms of y

1. ## how to expres in terms of y

dy/dt = t/2y+1. Find the general solution.

I get that y^2 +y = 1/2t^2 + C. How do I solve for Y?

2. Originally Posted by lord12
dy/dt = t/2y+1. Find the general solution.

I get that y^2 +y = 1/2t^2 + C. How do I solve for Y?
Complete the square.

$\displaystyle y^2 + y + \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{1}{2}t^2 + C$

$\displaystyle \left(y + \frac{1}{2}\right)^2 - \frac{1}{4} = \frac{1}{2}t^2 + C$

$\displaystyle \left(y + \frac{1}{2}\right)^2 = \frac{1}{2}t^2 + \frac{1}{4} + C$

$\displaystyle y + \frac{1}{2} = \sqrt{\frac{1}{2}t^2 + \frac{1}{4} +C}$

$\displaystyle y= \sqrt{\frac{1}{2}t^2 + \frac{1}{4} +C} - \frac{1}{2}$.

3. Originally Posted by Prove It
[snip]

$\displaystyle y + \frac{1}{2} = {\color{red}\pm} \sqrt{\frac{1}{2}t^2 + \frac{1}{4} +C}$

$\displaystyle y= {\color{red}\pm} \sqrt{\frac{1}{2}t^2 + \frac{1}{4} +C} - \frac{1}{2}$.
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4. Originally Posted by mr fantastic
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Yes - I was hoping that would be implied...

5. ## how can i tell if solution is unique

dy/dt = t/2y+1

y = sqrt(1/2t^2 + 1/4 +k)

y(1) = 3/2

6. Originally Posted by lord12
dy/dt = t/2y+1

y = sqrt(1/2t^2 + 1/4 +k)

y(1) = 3/2
First, your equation for $\displaystyle y$ is incorrect.

$\displaystyle y = \pm\sqrt{\frac{1}{2}t^2 + \frac{1}{4} + C} - \frac{1}{2}$.

If $\displaystyle y(1) = \frac{3}{2}$, then

$\displaystyle \frac{3}{2} = \pm \sqrt{\frac{1}{2}(1)^2 + \frac{1}{4} +C} - \frac{1}{2}$

$\displaystyle 2 = \pm \sqrt{\frac{1}{2} + \frac{1}{4} + C}$

$\displaystyle 4 = \frac{3}{4} + C$

$\displaystyle \frac{13}{4} = C$.

I think you'll find the solution is NOT unique...

$\displaystyle y = \pm \sqrt{\frac{1}{2}t^2+ \frac{1}{4} + \frac{13}{4}} - \frac{1}{2}$

$\displaystyle = \pm \sqrt{\frac{1}{2}t^2 + \frac{7}{2}} - \frac{1}{2}$.