Solving a Second Order Differential Equation: Help Needed

In summary, The given differential equation is of second order with both dependent and independent variables present. The coefficients are not constant, with one being a polynomial and the other a transcendental function. It is possible to find an analytical solution using series method, which is commonly introduced in first course differential equation textbooks. However, the book or instructor may not have provided enough explanation on the reasoning behind using this method. It is suggested to refer to an elementary differential equation book for a more detailed explanation on the series method.
  • #1
EvLer
458
0
Hello, not sure if it's a typo in the book but I can't work this out:

y'' + y(x^2 + e^x) = 0;

It's second order but both dependent and independent variables are present, and i am stuck.
You don't have to solve it for me entirely, a hint would be great.

Thanks in advance.
 
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  • #2
The coeff are not constant and one is a polynomial and the other a transcendental function.I doubt one can find an analytical method to find the 2 solutions.

Daniel.
 
  • #3
Are you saying that there's a typo in the problem?
(it's a first Diff Eq course)
 
  • #4
Not necessarily. Many first course d.e. texts introduce "series solutions" but that's the only way I see to do this.
 
  • #5
When in doubt guess. I ain't proud. I used NDSolve with initial conditions (just out of thin air) of y(0)=1,y'(0)=0. The results are below. I know it's not a solution but at least it's the start of a handle. :smile:

Edit: Wait a minute. That IS a solution. I mean it's not an analytically derived one. :smile:
 

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  • #6
HallsofIvy said:
Not necessarily. Many first course d.e. texts introduce "series solutions" but that's the only way I see to do this.
I'm sure it's somehow deducible with series, but unfortunately the book does not discuss that anywhere, i guess i'll have to google. I wish the book or the instructor would give more of "and why do we do this?" rather than "here's how: plug and chug".

Thanks everyone.
 
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  • #7
You should use series method to solve this diff eq.
It's hard to explain the method in this tiny box(!). I suggest to check out an elemantary diff eq book and read the chapter about the series method.
 
The 5 most frequently asked questions about "Solving a Second Order Differential Equation: Help Needed" are:

What is a second order differential equation?

A second order differential equation is a mathematical equation that involves the second derivative of an unknown function. It is commonly used to model physical systems in physics and engineering.

How do I know if a differential equation is second order?

A differential equation is considered second order if the highest derivative present in the equation is the second derivative. It can be written in the form of y''=f(x,y,y').

What are the steps for solving a second order differential equation?

The steps for solving a second order differential equation are: 1. Identify the order of the equation 2. Rewrite the equation in standard form 3. Determine the complementary function 4. Determine the particular integral 5. Combine the complementary function and particular integral to find the general solution 6. Apply initial conditions to find the particular solution.

Can I use any method to solve a second order differential equation?

Yes, there are several methods for solving a second order differential equation, such as the method of undetermined coefficients, variation of parameters, and the Laplace transform method. The choice of method will depend on the specific equation and initial conditions.

Are there any common pitfalls when solving a second order differential equation?

Some common pitfalls when solving a second order differential equation include: forgetting to check for a constant solution, forgetting to apply initial conditions, and making algebraic errors. It is important to double check your work and follow the steps carefully to avoid these mistakes.

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