By solving the initial value problem

$y^{\text{'}}=2t{y}^2,y(0)=y_0$

one can conlude that

$($Select all that apply$)$

For $y_0>0,$ solutions exist for all $t$

For solutions exist for all $t$

For solutions exist as long as

For $y_0>0,$ solutions exist as long as $t^2<mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac>$