Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 16 (a)

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MATH101 C April 2024

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Other MATH101 C Exams

• April 2024 •

Question 16 (a)
Consider the separable differential equation ${\frac {\mathrm {d} y}{\mathrm {d} x}}-2xy+4y=8-4x,\quad -\infty <x<+\infty .$ (a) Find the general solution. Express your answer in the form $y=f(x).$

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
Try to re-arrange the differential equation so that all terms involving $y$ are on the left, and all terms involving $x$ are on the right.

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.
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Solution
Re-arranging the differential equation yields ${\begin{aligned}{\frac {\mathrm {d} y}{\mathrm {d} x}}&=2xy-4x-4y+8=2(x-2)(y-2).\end{aligned}}$ Separating, ${\begin{aligned}{\frac {1}{y-2}}\;\mathrm {d} y&=(2x-4)\;\mathrm {d} x.\end{aligned}}$ Taking the integral, ${\begin{aligned}\int {\frac {1}{y-2}}\;\mathrm {d} y&=\int (2x-4)\;\mathrm {d} x.\\\log \|y-2\|&=x^{2}-4x+c.\\\|y-2\|&=e^{x^{2}-4x+c}=Ce^{x^{2}-4x},\end{aligned}}$ where $C=e^{c}$ . To get rid of the absolute value, note that if $y-2$ is negative, then we can replace $C$ with $-C$ . Therefore, for some constant $C$ , we have ${\begin{aligned}y-2=Ce^{x^{2}-4x},\end{aligned}}$ which simplifies to ${\begin{aligned}y=2+Ce^{x^{2}-4x}.\end{aligned}}$