The image is a close-up, top-down view of a page from a mathematics exam paper. The paper is white and has faint horizontal lines from a previous imprint. The exam questions are printed in black ink, with some handwritten annotations and markings in blue and purple ink. The visible questions are from "Question One," "Question Two," and "Question Three." **Question One** has three parts: * Part (a) asks to "State the Existence and Uniqueness theorem." * Part (b) asks to "Given the domain D of all points satisfying $|x-0|\leq1, |y-1|\leq1$, on what interval would the initial value problem $\frac{dy}{dx} = y$ have a unique solution." A blue checkmark is present next to this part. * Part (c) asks to "Define Laplace transform of a function $y(t)$ and hence, Using Laplace transform method solve $y''(t)-y'(t)+y(t)=0, y(0)=0, y'(0)=1, y''(0)=2$." **Question Two** has three parts: * Part (a) asks to "Define a differential equation and state the types." * Part (b) asks to "Use D-Operators method to solve $y''(x)+y'(x)-2y(x)=4x$." * Part (c) asks to "Use variation of parameters method to solve $y''(x)-y'(x)-6y(x)=e^{-2x}$." **Question Three** has three parts: * Part (a) asks to "Solve the differential equations (i) $(x^2-9)\frac{dy}{dx} + xy = 0$ (ii) $(e^{2^y} - y) \cos x \frac{dy}{dx} = e^y \sin 2x$." * Part (b) asks to "Find the general solution to $(x^2-9)\frac{dy}{dx} + xy = 0$." A blue checkmark is present next to this part. * Part (c) asks to "Solve $\frac{dy}{dx} + y = x$, $y(0)=4$." There are also handwritten notes in purple ink at the bottom right and bottom center of the image, which appear to be working or potential answers related to the domain $|x-0|\leq1$ and $|y-1|\leq1$ from Question One (b), with inequalities such as $-0 \leq x \leq 1$ and $y=2$. A blue scribble is also present near part (b) of Question Three. The background is a plain, textured white surface, typical of paper. There is no discernible environment or location other than that of an academic document.

$The image is a close-up, top-down view of a page from a mathematics exam paper. The paper is white and has faint horizontal lines from a previous imprint. The exam questions are printed in black ink, with some handwritten annotations and markings in blue and purple ink. The visible questions are from "Question One," "Question Two," and "Question Three." **Question One** has three parts: * Part (a) asks to "State the Existence and Uniqueness theorem." * Part (b) asks to "Given the domain D of all points satisfying $|x-0|\leq1, |y-1|\leq1$, on what interval would the initial value problem $\frac{dy}{dx} = y$ have a unique solution." A blue checkmark is present next to this part. * Part (c) asks to "Define Laplace transform of a function $y(t)$ and hence, Using Laplace transform method solve $y''(t)-y'(t)+y(t)=0, y(0)=0, y'(0)=1, y''(0)=2$." **Question Two** has three parts: * Part (a) asks to "Define a differential equation and state the types." * Part (b) asks to "Use D-Operators method to solve $y''(x)+y'(x)-2y(x)=4x$." * Part (c) asks to "Use variation of parameters method to solve $y''(x)-y'(x)-6y(x)=e^{-2x}$." **Question Three** has three parts: * Part (a) asks to "Solve the differential equations (i) $(x^2-9)\frac{dy}{dx} + xy = 0$ (ii) $(e^{2^y} - y) \cos x \frac{dy}{dx} = e^y \sin 2x$." * Part (b) asks to "Find the general solution to $(x^2-9)\frac{dy}{dx} + xy = 0$." A blue checkmark is present next to this part. * Part (c) asks to "Solve $\frac{dy}{dx} + y = x$, $y(0)=4$." There are also handwritten notes in purple ink at the bottom right and bottom center of the image, which appear to be working or potential answers related to the domain $|x-0|\leq1$ and $|y-1|\leq1$ from Question One (b), with inequalities such as $-0 \leq x \leq 1$ and $y=2$. A blue scribble is also present near part (b) of Question Three. The background is a plain, textured white surface, typical of paper. There is no discernible environment or location other than that of an academic document.$ Study materials and everyday objects, sh

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Sat, Jun 20, 2026, 05:16 PM

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The image is a close-up, top-down view of a page from a mathematics exam paper. The paper is white and has faint horizontal lines from a previous imprint. The exam questions are printed in black ink,…read more

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