# Differential equation x y' +x^2 y'' = k^2 y

• ehrenfest
In summary, the conversation discusses a differential equation involving the variable x and a constant k. The question is how to prove that the set of solutions {x^r}, where r is a real number, forms a basis for the solution space of the equation. The conversation goes on to suggest using a substitution and solving for x in order to show that the set of solutions spans the entire solution space.
ehrenfest
[SOLVED] differential equation

## Homework Statement

x y' +x^2 y'' = k^2 y

where y=y(x), k is constant.

How do you prove that x^r, where r is a real number form a basis for that differential equation? They are obviously linearly independent. But how do you prove that they span the solution space?

## The Attempt at a Solution

By solving it!
Try the substitution $y(x)=u(x)\,x^k$

i didn't get what exactly you want to prove! do you want to solve it for x?

When I do that I get (2k+1)u'(x)+x u''(x)=0. Is that a contradiction?

No it isn't! Now let $u'(x)=a(x),\,u''(x)=a'(x)$ which yields a separable 1st order ODE.

astrosona said:
i didn't get what exactly you want to prove! do you want to solve it for x?

I want to find out if the set of solutions {x^r}, $$r \in \mathbb{R}$$ spans the entire solution space of that equation.

Then I get $$a(x)=\frac{C}{x^{2k+1}}$$, where C is a constant. Very nice Rainbow Child!

oh... i see, thank you

## 1. What is a differential equation?

A differential equation is an equation that relates a function and its derivatives. It describes the relationship between a function and its rate of change.

## 2. What is the order of this differential equation?

The order of a differential equation is determined by the highest derivative present. In this equation, the highest derivative is y'', so it is a second-order differential equation.

## 3. What is the meaning of the constants k and x in this equation?

The constant k represents a fixed value in the equation, while x represents the independent variable. The values of these constants can affect the behavior of the solution to the differential equation.

## 4. How can this differential equation be solved?

There are various methods for solving differential equations, such as separation of variables, integrating factors, and using substitution. The specific method used to solve this equation would depend on the initial conditions and the type of differential equation it is (e.g. linear, non-linear).

## 5. What are the applications of this type of differential equation?

Differential equations are used to model many natural phenomena and physical systems, such as population growth, chemical reactions, and electrical circuits. In this specific equation, it could be used to model a damped harmonic oscillator or a vibrating string.

• Calculus and Beyond Homework Help
Replies
7
Views
904
• Calculus and Beyond Homework Help
Replies
27
Views
665
• Calculus and Beyond Homework Help
Replies
3
Views
522
• Calculus and Beyond Homework Help
Replies
10
Views
692
• Calculus and Beyond Homework Help
Replies
7
Views
751
• Calculus and Beyond Homework Help
Replies
18
Views
2K
• Calculus and Beyond Homework Help
Replies
7
Views
636
• Calculus and Beyond Homework Help
Replies
2
Views
414
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
1K