Solving an Ordinary Differential Equation Problem: x*x''-y*y''=0

That makes me think of sine and cosine.In summary, the conversation discussed solving a problem involving a system of differential equations, specifically x*x''-y*y''=0 and x''+y''+x+y=0, which was published in Ince's book "Ordinary Differential Equations" in 1909. The participants also mentioned assuming x and y are twice differentiable functions and being interested in the methods used by others. One participant suggested using the relationship yy"=xx" and the other participant suggested thinking of sine and cosine as a solution.
  • #1
lurflurf
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I have been looking though Ince (Ordinary Differential Equations), nice book.
I enjoyed solving this problem:
6) pp. 157
Integrate the system
x*x''-y*y''=0
x''+y''+x+y=0
[Edinburgh, 1909.]
we may assume x and y are twice differentiable functions
defined everywhere from reals to reals
' denotes differentiation

I would be interested in the methods used by others.
 
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  • #2
lurflurf said:
I have been looking though Ince (Ordinary Differential Equations), nice book.
I enjoyed solving this problem:
6) pp. 157
Integrate the system
x*x''-y*y''=0
x''+y''+x+y=0
[Edinburgh, 1909.]
we may assume x and y are twice differentiable functions
defined everywhere from reals to reals
' denotes differentiation

I would be interested in the methods used by others.
Well, my first reaction would be that yy"= xx" so x"= (y/x)y".
Then x"+ y"+ x+ y= 0 becomes (y/x)y"+ y"= (y+x)y"/x= -(x+y) so y"= -x. Put that back into the original equation and we have x"= y. Differentiating twice more gives xIV= -x.
 
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FAQ: Solving an Ordinary Differential Equation Problem: x*x''-y*y''=0

What is an ordinary differential equation (ODE)?

An ordinary differential equation is a type of mathematical equation that describes the relationship between a function and its derivatives. It involves one independent variable and one or more dependent variables.

What is the general form of an ODE?

The general form of an ODE is dy/dx = f(x,y), where y is the dependent variable, x is the independent variable, and f(x,y) is a function that relates the two variables.

What is the purpose of solving an ODE problem?

The purpose of solving an ODE problem is to find a function that satisfies the given equation and its initial conditions. This function can then be used to make predictions or solve real-world problems.

How do you solve an ODE problem?

To solve an ODE problem, you can use various methods such as separation of variables, substitution, or integrating factors. These methods involve manipulating the equation to isolate the dependent variable and integrating to find the function that satisfies it.

How is the ODE problem x*x''-y*y''=0 solved?

The ODE problem x*x''-y*y''=0 can be solved by first rewriting it in the form of dy/dx = f(x,y). Then, using the substitution y = vx, the equation can be transformed into a separable form. After integrating both sides, the function y(x) can be found by solving for v and substituting back in the original equation.

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