<exercise masterit-seed="2307" masterit-slug="F4" masterit-name="Implicit solutions for exact IVPs">
<statement>
<p>
Determine which of the following ODEs is exact.
</p>
<me> -9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y </me>
<me> -9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, 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> -9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y </me>
<p>
Its implicit solution satisfying the initial value is:
</p>
<me> -4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right) </me>
</answer>
</exercise>
