<exercise masterit-seed="9233" 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^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'} </me>
<me> -8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 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> -8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y </me>
<p>
Its implicit solution satisfying the initial value is:
</p>
<me> 2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right) </me>
</answer>
</exercise>
