<exercise masterit-seed="9506" masterit-slug="F4" masterit-name="Implicit solutions for exact IVPs">
<statement>
<p>
Determine which of the following ODEs is exact.
</p>
<me> -4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y </me>
<me> -4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, 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> -4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2} </me>
<p>
Its implicit solution satisfying the initial value is:
</p>
<me> y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right) </me>
</answer>
</exercise>
