A researcher has conducted an experiment concerning the placement of an object, given as y in centimetres, at time t in hours, after an initial force on 22/02/2023 at 9:00 a.m. disturbed the object from rest. It is a particularly delicate experiment and only the following data is gathered successfully. Even with such limited data, to advance on the project the researcher considers, separately, the following three possible models for the experiment. Each model consists of a differential equation. dtdy=6tet2dtdy=2ydt2d2y+4dtdy=5y 1. Find the general solution y(t) to model (1). Then give the particular solution that satisfies y being 3.2cm from an initial reference point at the very start of the experiment. 2. The solutions y(t)=Ae2t and y(t)=e2t(Acos(t)+5sin(t) ) (where A is an arbitrary constant) solve models (2) and (3), but the researcher cannot remember which one solves which. (a) Find which solution solves which model, clearly showing your working. (b) Give the particular solution for each model that satisfies y being 3.2cm from an initial reference point at the very start of the experiment. 3. Accepting that y is 3.2cm from an initial reference point at the very start of the experiment, construct a table mirroring the one above, where the values for y,Y and its rate of change, are the values given by model (2). Then, produce the same table that fits model (3). Give results to 2 significant figures. 4. Construct a table of the differences in the recorded and predicted y values and the recorded and predicted rates of change of the y values for both model (2) and (3). Give results to a single significant figure. Does a particular model appear to fit the data better and if so, which one and why? Your explanation should be short, simply showing your understanding. A single sentence of around 1020 words is all that is required for clarity. 5. Feeling curious, the researcher then considers the following model for the experiment, accepting y as 3.2cm from an initial reference point at the very start of the experiment: dtdy=2y+sin(t). A researcher has conducted an experiment concerning the placement of an object, given as y in centimetres, at time t in hours, after an initial force on 22/02/2023 at 9:00 a.m. disturbed the object from rest. It is a particularly delicate experiment and only the following data is gathered successfully. Even with such limited data, to advance on the project the researcher considers, separately, the following three possible models for the experiment. Each model consists of a differential equation. dtdy=6tet2dtdy=2ydt2d2y+4dtdy=5y 1. Find the general solution y(t) to model (1). Then give the particular solution that satisfies y being 3.2cm from an initial reference point at the very start of the experiment. 2. The solutions y(t)=Ae2t and y(t)=e2t(Acos(t)+5sin(t) ) (where A is an arbitrary constant) solve models (2) and (3), but the researcher cannot remember which one solves which. (a) Find which solution solves which model, clearly showing your working. (b) Give the particular solution for each model that satisfies y being 3.2cm from an initial reference point at the very start of the experiment. 3. Accepting that y is 3.2cm from an initial reference point at the very start of the experiment, construct a table mirroring the one above, where the values for y,Y and its rate of change, are the values given by model (2). Then, produce the same table that fits model (3). Give results to 2 significant figures. 4. Construct a table of the differences in the recorded and predicted y values and the recorded and predicted rates of change of the y values for both model (2) and (3). Give results to a single significant figure. Does a particular model appear to fit the data better and if so, which one and why? Your explanation should be short, simply showing your understanding. A single sentence of around 1020 words is all that is required for clarity. 5. Feeling curious, the researcher then considers the following model for the experiment, accepting y as 3.2cm from an initial reference point at the very start of the experiment: dtdy=2y+sin(t)
