It looks like you've uploaded a PDF document. Let me review it and extract the relevant questions to assist you in solving them. PyPDF2 import PdfReader

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pdf_path = '/mnt/data/1739298372452kvyqd0vj.pdf' reader = PdfReader(pdf_path)

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pdf_text = "" for page in reader.pages: pdf_text += page.extract_text()

Display the first 1000 characters to understand the content

pdf_text[:1000]It appears that the document contains several differential equations-related problems. Here are the problems outlined in the text:

1. Problem 1: Determine if the function $y = \frac{e^{3x} + 2}{3e^{2x}}$ is a solution to the differential equation: $y'' + 5y' + 6y = 0$ If it is, find all possible intervals $I$ for which it is a solution.

2. Problem 2: Determine if the equation $x^2 + t^2 - 6t + 10x - 24 = 147$ is a solution to the differential equation: $(x + 5)x' = 3 + t$ If it is, find all possible intervals $I$ for which it is a solution.

3. Problem 3: Consider the IVP (Initial Value Problem): $y' = \left( y^2 \right) \quad \text{with initial conditions} \quad y(1) = 2, \, y(0) = 1.$ (a) Show that the Existence Theorem does not guarantee a solution to this IVP. (b) Show that $y = \left( 1.5x \right)^2 + 3$ is a solution to the IVP on some interval $I$. Does this result contradict your work in part (a)?

4. Problem 4: Consider the IVP: $y' = (2y^3) \cos(xy), \quad y(0) = 1.$

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