1. \( \frac{d y}{d x}=\frac{x}{y} \) and \( y=2 \) when \( x=1 \) 2. \( \frac{d y}{d x}=-\frac{x}{y} \) and \( y=3 \) when \( x=4 \) 3. \( \frac{d y}{d x}=\frac{y}{x} \) and \( y=2 \) when \( x=2 \) 4. \( \frac{d y}{d x}=2 x y \) and \( y=3 \) when \( x=0 \) 5. \( \frac{d y}{d x}=(y+5)(x+2) \) and \( y=1 \) when \( x=0 \) 6. \( \frac{d y}{d x}=\cos ^{2} y \) and \( y=0 \) when \( x=0 \) 7. \( \frac{d y}{d x}=\left(\cos ^{2}\right) e^{y+\sin x} \) and \( y=0 \) when \( x=0 \) 8. \( \frac{d y}{d x}=e^{x-y} \) and \( y=2 \) when \( x=0 \) 9. \( \frac{d y}{d x}=-2 x y^{2} \) and \( y=0.25 \) when \( x=1 \) 10. \( \frac{d y}{d x}=\frac{4 \sqrt{y} \ln x}{x} \) and \( y=1 \) when \( x=e \)

The study of differential equations has a rich history that dates back to the 17th century, with pioneers such as Isaac Newton and Gottfried Wilhelm Leibniz laying down the foundational concepts. These equations serve as a powerful tool for modeling a variety of natural phenomena, from population growth to the motion of celestial bodies. Understanding their solutions not only helps in theoretical exploration but also in practical application across many scientific fields. One of the most common real-world applications of differential equations is in the field of engineering. For instance, they are crucial in designing systems such as electrical circuits, where the relationship between voltage, current, and resistance can be described using differential equations. Similarly, in biology, they model populations or the spread of diseases, helping researchers and public health officials make informed decisions to combat outbreaks. The versatility of these equations truly makes them indispensable in numerous aspects of life and science!