Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 21

MATH100 A December 2023

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Question 21
Possible solutions to a differential equation Which of the following is a solution to the differential equation $y'={\frac {y}{2}}$ with initial condition $y(0)=1$? Select all that apply.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The initial condition $y(0)=1$ tells us that the derivative $y'(0)={\frac {1}{2}}$, which is a positive number. What does this mean in terms of the growth of the function $y$?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The initial condition $y(0)=1$ rules out the top-left, top-right and bottom-centre graphs. The fact that the slope of the tangent line at $x=0$ is ${\frac {1}{2}}$ rules out the top-centre and bottom-left graphs, which have tangent lines of slope $\geq 1$, so only the bottom-right graph can be a solution of the differential equation. Alternatively, the solutions of $y'=ky$ are of the form $y(x)=Ce^{kx}$, where $C$ is a constant. In this case, the solution is $y(x)=e^{x/2}$, and we can use this function to rule out all of the graphs, save for the bottom-right one.