Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on the following concepts:

• Homogeneous functions and their definition via scaling property
• Degree of homogeneity ($n$ where $f(tx,ty)=t^{n}f(x,y)$)
• Homogeneous differential equations (first-order)
• Substitution method for solving homogeneous ODEs ($v=\frac{y}{x}$ or $v=\frac{x}{y}$)
• Transformation of homogeneous ODEs into separable form
• Initial value problems (IVP) involving homogeneous differential equations
• Integration techniques after variable substitution

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Important Formulas

Below are the key formulas used in this exercise:

Homogeneous Function Definition: $f(tx,ty)=t^{n}f(x,y)$ where $n$ is the degree of homogeneity.

Substitution for Homogeneous ODEs: When $y=vx$ (where $v=\frac{y}{x}$): $\frac{dy}{dx}=v+x\frac{dv}{dx}$

When $x=vy$ (where $v=\frac{x}{y}$): $\frac{dx}{dy}=v+y\frac{dv}{dy}$

General Form of Homogeneous ODE: $\frac{dy}{dx}=\frac{f_1(x,y)}{f_2(x,y)}$ where $f_1$ and $f_2$ are homogeneous functions of the same degree.

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Summary

This exercise covers the identification of homogeneous functions using the scaling property $f(tx,ty)=t^{n}f(x,y)$ to determine the degree $n$. It progresses to solving homogeneous differential equations through the substitution $y=vx$ (or $x=vy$), which reduces the equation to separable variables. The exercise emphasizes algebraic manipulation to recognize homogeneous forms, proper application of substitutions, and handling initial value problems by determining constants of integration from given conditions. Common patterns include rational functions of $\frac{y}{x}$ and terms involving $\sqrt{xy}$ or $e^{y/x}$.