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[Solved] (a) An equation that contains a function as well as its derivative is known as a differential equation. The most basic form for a differential

Question: (a) An equation that contains a function as well as its derivative is known as a differential equation. The most basic form for a differential equation is given by:

\[\frac{d y}{d x}=k y\]

where \(y\) is some function of \(x\)

The equation above is equivalent to the statement: "The height of the function \(y\), is proportional to its gradient at any given point". Since this fits the description of an exponential function, it suggests that a possible solution to this differential equation is given by \(y=e^{k x}\).

Show that \(y=e^{k x}\) satisfies the differential equation above.

(b) A more complicated differential equation involves the second derivative:

\[\frac{d^{2} y}{d x^{2}}=-k^{2} y\]
  1. By considering the various functions studied throughout this course, find a function that satisfies this differential equation. Justify your answer with calculations. (3 marks)
  2. Find a second function that also satisfies this equation. ( 1 mark)
  3. How many possible solutions to this differential equations are there? Discuss. ( 4 marks)

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