Convert $\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$ and $\ddot{y}(t) = \dot{y}(t) -y(t)^2$ into 1. Order IVP.

Question

1) We're given the IVP 3. Order $$\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$$ with initial values $y(0)=a, \dot{y}(0)=b, \ddot{y}(0)=c$ and want to convert it into a 1. Order IVP.

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2) We're given the IVP 2. Order $$\ddot{y}(t) = \dot{y}(t) -y(t)^2$$ with initial values $$y(0)=y_0, \dot{y}(0)=y_1$$ and want to convert it into a 1. Order IVP.

Answer 1

1) We let $$ (z_0(t), z_1(t), z_2(t))^T = (y(t), \dot{y}(t), \ddot{y}(t))^T $$ Thus $$ \frac{dz}{dt}(t)= (z_1(t), z_2(t), (z_1(t) - z_0(t))^2 + 3 \sin{(t)}z_0(t))^T $$ with $$ z(0)=(a, b, c)^T $$ is our solution.

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2) Again, we let $$(z_0(t), z_1(t))^T = (y(t), \dot{y}(t))^T$$ Thus $$\frac{dz}{dt}(t) = (z_1(t), z_1(t)-z_0(t)^2)^T$$ with $$z(0)= (y_0, y_1)^T$$ is our solution.