- #1

Dell

- 590

- 0

for 4.1)

y=e

^{x}=y' =y''

xy'' - (x+1)y' + y = x

^{2}

xe

^{x}- (x+1)e

^{x}+ e

^{x}= 0

none of these seem to work, what am i doing wrong?

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- Thread starter Dell
- Start date

In summary, the task at hand is to show that y = e^x and y = x + 1 are solutions to the homogeneous equation xy'' - (x+1)y' + y = 0, and that they are basic solutions, meaning they are linearly independent and span the space of solutions. This is important because these basic solutions can be used to find all other solutions to the homogeneous equation. The given equations are not homogeneous, but by dropping the terms on the right side, they can be made into homogeneous equations. The solutions y1 and y2 should be used separately, but any linear combination of them is also a solution.

- #1

Dell

- 590

- 0

for 4.1)

y=e

xy'' - (x+1)y' + y = x

xe

none of these seem to work, what am i doing wrong?

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- #2

Dell

- 590

- 0

are these equations even homogeneous?

- #3

Mark44

Mentor

- 37,753

- 10,101

For 4.1, the homogeneous equation is xy'' - (x + 1)y' + y = 0. It's easy to show that y_1 and y_2 are solutions to this homogeneous equation. By "basic" I think it means that y_1 and y_2 are linearly independent (easy to show) and span the space of solutions to the homogeneous equation, which is of dimension 2.

Same approach for 4.2.

- #4

Dell

- 590

- 0

- #5

Mark44

Mentor

- 37,753

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Don't overthink this. You are not being asked to find solutions - just verify that the given functions are solutions to the homogeneous equations.

- #6

Dell

- 590

- 0

also am i supposed to use y1 and y2 seperately or at the same time, ie use y1 once prove that it is a solution and then use y2. or am i meant to use them together somehow

- #7

Mark44

Mentor

- 37,753

- 10,101

I guess because they are assuming you know the difference between a nonhomogeneous equation and the associated homogeneous equation.Dell said:but i don't understand why they would give me the nonhomogeneous equation

Separately. However, if yDell said:also am i supposed to use y1 and y2 seperately or at the same time, ie use y1 once prove that it is a solution and then use y2. or am i meant to use them together somehow

A homogeneous differential equation is a type of differential equation where the dependent variable and its derivatives appear in terms of a single independent variable, and all the terms in the equation have the same degree. This means that all the terms can be expressed as a multiple of the dependent variable and its derivatives.

The general approach to solving a homogeneous differential equation is by using separation of variables. This involves separating the dependent and independent variables on opposite sides of the equation and then integrating both sides to solve for the dependent variable. In some cases, substitution or using a special integrating factor may also be necessary.

The main difference between a homogeneous and non-homogeneous differential equation is that in a homogeneous equation, all terms have the same degree, whereas in a non-homogeneous equation, the degrees of the terms may vary. This means that the methods used to solve these two types of equations may differ.

No, there are some differential equations that cannot be classified as either homogeneous or non-homogeneous. These are known as partial differential equations, which involve partial derivatives of multiple variables. These equations require more advanced techniques to solve.

Homogeneous differential equations have many applications in various fields of science and engineering. They are commonly used in physics to model the behavior of physical systems, in chemistry to study reaction kinetics, and in biology to understand population dynamics. They are also used in engineering for designing control systems and in economics for studying market dynamics.

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