<exercise masterit-seed="6299" masterit-slug="F4" masterit-name="Implicit solutions for exact IVPs">
<statement>
<p>
Determine which of the following ODEs is exact.
</p>
<me> 6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3} </me>
<me> 6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2} </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> 6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3} </me>
<p>
Its implicit solution satisfying the initial value is:
</p>
<me> -t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5 </me>
</answer>
</exercise>
