<exercise masterit-seed="6141" masterit-slug="F4" masterit-name="Implicit solutions for exact IVPs">
<statement>
<p>
Determine which of the following ODEs is exact.
</p>
<me> 3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'} </me>
<me> 12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t 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> 3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'} </me>
<p>
Its implicit solution satisfying the initial value is:
</p>
<me> -y^{4} + 3 \, t^{2} y + 2 \, t y = 0 </me>
</answer>
</exercise>
