<exercise masterit-seed="0493" masterit-slug="F4" masterit-name="Implicit solutions for exact IVPs">
<statement>
<p>
Determine which of the following ODEs is exact.
</p>
<me> -4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t </me>
<me> -4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'} </me>
<p>
Then find an implicit solution for this exact ODE
satisfying the initial value
<m>y( 1 )= -1 </m>.
</p>
</statement>
<answer>
<p>The following ODE is exact.</p>
<me> -4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t </me>
<p>
Its implicit solution satisfying the initial value is:
</p>
<me> 3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12 </me>
</answer>
</exercise>
