2nd ODE constant coeff, quick question

In summary, the problem involves solving a second-order differential equation with a particular and complementary solution. The complementary solution is found to be y=C_1e^{-x} and the particular solution is of the form y_{PI}=Ax^3+Bx^2+Cx+D. While A, B, and C can be determined by differentiating and substituting into the equation, the value of D is redundant since it can be determined by the initial conditions rather than the forcing function. Another method to find D is by integrating the equation and using the integrating factor.
  • #1
rock.freak667
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Homework Statement


Solve:
[tex]\frac{d^2y}{dx^2}+\frac{dy}{dx}=3x^2+2x+1[/tex]

The Attempt at a Solution



Well the C.F. is [itex]y=C_1e^{-x}[/itex]
the P.I. is of the form [itex]y_{PI}=Ax^3+Bx^2+Cx+D[/itex]

I can find the values of A,B and C bu differentiating it and substituting it into the equation. But How would I find D since there is no 'y' in the ODE given and differentiating [itex]y_{PI}[/itex] makes the constant disappear.
(Note: I can find the answer by integrating it w.r.t x and then using the integrating factor but I would like to know how to find it by adding the PI and CF together)
 
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  • #2
rock.freak667 said:

Homework Statement


Solve:
[tex]\frac{d^2y}{dx^2}+\frac{dy}{dx}=3x^2+2x+1[/tex]

The Attempt at a Solution



Well the C.F. is [itex]y=C_1e^{-x}[/itex]
the P.I. is of the form [itex]y_{PI}=Ax^3+Bx^2+Cx+D[/itex]

I can find the values of A,B and C bu differentiating it and substituting it into the equation. But How would I find D since there is no 'y' in the ODE given and differentiating [itex]y_{PI}[/itex] makes the constant disappear.



(Note: I can find the answer by integrating it w.r.t x and then using the integrating factor but I would like to know how to find it by adding the PI and CF together)

The D is redundant since one of the solutions to the natural equation is a constant function.

Therefor the constant function is determined by the initial conditions rather then the forcing function.
 
  • #3
Ah...thanks then,I thought I was really doing something wrong.
 

Related to 2nd ODE constant coeff, quick question

What is a 2nd order ODE with constant coefficients?

A 2nd order ordinary differential equation (ODE) with constant coefficients is an equation in which the highest derivative term has a constant coefficient, and the independent variable appears only in first and second derivatives.

What is the general form of a 2nd order ODE with constant coefficients?

The general form of a 2nd order ODE with constant coefficients is y'' + ay' + by = f(x), where a and b are constants and f(x) is a function of x.

How do you solve a 2nd order ODE with constant coefficients?

To solve a 2nd order ODE with constant coefficients, you can use the method of undetermined coefficients, variation of parameters, or the method of Laplace transforms.

What are the initial conditions for a 2nd order ODE with constant coefficients?

The initial conditions for a 2nd order ODE with constant coefficients are two values of the dependent variable (usually denoted as y) and its first derivative (usually denoted as y') at a specific point (usually denoted as x = x0).

Can a 2nd order ODE with constant coefficients have complex solutions?

Yes, a 2nd order ODE with constant coefficients can have complex solutions. This occurs when the roots of the characteristic equation (ay^2 + by + c = 0) are complex numbers.

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