Solving a 2nd Order Differential Equation with Auxiliary Method

In summary, the conversation discusses solving an equation using the standard auxiliary method. The speaker shares their auxiliary equation and particular solution, and questions if their approach is correct. They also mention using the Laplace method and getting a different answer. They seek help in confirming their solution and mention the possibility of a mistake in their auxiliary equation. Finally, they receive a suggested solution from another speaker.
  • #1
angel
18
0
ok
im trying to solve the following equation using standard aux method:

d^2y/dx^2 + 3dy/dy +2y = cos x with conditions x(0)=-3 and x'(0) = 3

my aux eqn is:

ae^x + be^-2x

and my yp is;

a sin kx + b cos kx

i differentiate this twice and substitute into the original equation, and i get:

(-a sin x - b cos x)+3(a cos x-b sin x) + 2(a sin x+ b cos x) = cos x.

Now at this stage I am not sure if this is right or not, can someone please confirm if this is right or not.

I ve done this question using the laplace method and the answer i get is;

-19/2e^x +32/5 e^2x + 1/10 cos x - 3/10 sin x.

Here my aux equation is not the same as what i get when i apply the laplace method.

Im not sure what is going wrong.
Someone help me please.
 
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  • #2
you have a and b representing two different numbers simlutaneously.

if you're not sure which of e^2x or e^-2x is correct why don't you subs it into the equation to see which one works? i think it is -2, i also think you want e^-x and not e^x since the aux equation is m^2+3m+2=(m+1)(m+2)
 
  • #3
solution

Hi;
Here is your solution:
[tex]
y(x)=\frac{3sinx}{10}+\frac{cosx}{10}-\frac{7e^{-x}}{2}+\frac{2e^{-2x}}{5}
[/tex]
Best of luck,
Max.
P.S. Obtained by Maxima: http://maxima.sourceforge.net/download.shtml
 

1. What is a 2nd order differential equation?

A 2nd order differential equation is a mathematical equation that involves the second derivative of a function. It is commonly used to model physical phenomena in science and engineering.

2. What is the auxiliary method for solving a 2nd order differential equation?

The auxiliary method, also known as the substitution method, is a technique used to simplify a 2nd order differential equation into a first order differential equation. This is done by substituting a new variable for the dependent variable in the original equation.

3. What are the steps for solving a 2nd order differential equation using the auxiliary method?

The steps for solving a 2nd order differential equation with the auxiliary method are as follows:

  1. Identify the dependent variable and its derivatives in the equation.
  2. Substitute a new variable for the dependent variable.
  3. Find the first derivative of the new variable.
  4. Substitute the first derivative into the original equation.
  5. Solve the resulting first order differential equation.
  6. Substitute the original variable back in to get the final solution.

4. What are some common applications of 2nd order differential equations?

2nd order differential equations are used in many areas of science and engineering, including physics, chemistry, biology, economics, and engineering. They can be used to model motion, growth, decay, and many other physical phenomena.

5. Are there any limitations to the auxiliary method for solving 2nd order differential equations?

The auxiliary method is only applicable to certain types of 2nd order differential equations, specifically those that can be simplified into a first order equation. It may not work for more complex equations or those with non-constant coefficients. Other methods, such as the power series method, may be necessary for these cases.

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