To determine which of these differential equations are not homogeneous, we need to analyze each equation and see if it can be expressed in terms of homogeneous functions of the same degree.

Explanation of Homogeneous Differential Equations

A differential equation is considered homogeneous if it can be written in the form: $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$ or if both $M(x, y)$ and $N(x, y)$ in the general form $M(x, y)dx + N(x, y)dy = 0$ are homogeneous functions of the same degree.

Now, let’s go through each equation in the image:

1. $xy' = y - x$ - Check for homogeneity.
2. $xy' = y + x$ - Check for homogeneity.
3. $(x - y)dx + (x - y + 1)dy = 0$ - Likely not homogeneous due to the constant $+1$ term.
4. $y' = \frac{y^2 + x^2}{2xy}$ - Check for homogeneity.
5. $\frac{dy}{dx} = \frac{x}{y} + \frac{y}{x}$ - Check for homogeneity.
6. $\left( y + \sqrt{x^2 + y^2} \right) dx = xdy$ - Likely not homogeneous due to the square root term.
7. $x(x + y)dy = (x^2 + y^2)dy$ - Check for homogeneity.
8. $xy' - y = x^2 e^x$ - Likely not homogeneous due to $e^x$ term.
9. $xy' = x^2 \sin x + y$ - Likely not homogeneous due to $\sin x$ term.
10. $(y + x)y' = x - y$ - Check for homogeneity.
11. $(7x + 2y)y' = -2x - 7y$ - Check for homogeneity.
12. $(3y^2 + x^2)y' + 2xy + 3x^2 = 0$ - Likely not homogeneous due to mixed powers.
13. $(2xy + x^2 + 3y^2)y' + (y^2 + 2xy + 3x^2) = 0$ - Likely not homogeneous due to mixed powers.

Based on the analysis:

Non-Homogeneous Equations

• Equation 3: Contains a constant $+1$ term, breaking homogeneity.
• Equation 6: Contains a square root term $\sqrt{x^2 + y^2}$, which is not homogeneous.
• Equation 8: Contains an exponential term $e^x$, which is not homogeneous.
• Equation 9: Contains a trigonometric term $\sin x$, which is not homogeneous.
• Equation 12: Mixed terms with powers that likely break homogeneity.
• Equation 13: Mixed terms with powers that likely break homogeneity.

Thus, equations 3, 6, 8, 9, 12, and 13 are not homogeneous.

Would you like a more detailed explanation on any specific equation, or do you have any questions?

Further Exploration Questions

1. How can we determine the degree of homogeneity in a differential equation?
2. What are some examples of real-world applications where homogeneous differential equations are used?
3. Could equations with exponential or trigonometric terms ever be homogeneous?
4. What techniques are commonly used to solve non-homogeneous differential equations?
5. How does the presence of constants affect the homogeneity of a differential equation?

Tip:

For homogeneous equations, a common substitution is $v = \frac{y}{x}$ which simplifies the equation by reducing variables.