Second Order Differential Equation: y''-9y=4+5sinh(3x)

[lch]

Hi all,

Consider a homogeneous linear ODE with constant coefficient: y''-9y=4+5sinh(3x), and the homogeneous solution is given by y_h (x)= C₁e^3x+C₂e^-3x.
But I don't know how to find out the particular solution, and the answer is y_p(x)=-(4/9)+(5/6)x cosh(3x).

Sorry for my poor English and thanks for your help!

 

[Steven G]

Hi all,

Consider a homogeneous linear ODE with constant coefficient: y''-9y=4+5sinh(3x), and the homogeneous solution is given by y_h (x)= C₁e^3x+C₂e^-3x.
But I don't know how to find out the particular solution, and the answer is y_p(x)=-(4/9)+(5/6)x cosh(3x).

Sorry for my poor English and thanks for your help!

Your English is fine. What have you tried, where are you stuck? If we know where you are having trouble, then we know how to help you. So please show us your work, even if you know its wrong and you will get excellent help.

 

[lch]

Your English is fine. What have you tried, where are you stuck? If we know where you are having trouble, then we know how to help you. So please show us your work, even if you know its wrong and you will get excellent help.

Well, actually I have no idea about what to do next.
What I have tried is.....

m²-9=0
So, y_p=C₂x²+C₁x+C₀
y_p'= 2C₂x+C₁
y_p''=2C₂

Then by substituting back into the equation, I have got
2C₂-9(C₂x²+C₁x+C₀)= 4+5sinh(3x)

By comparing the like terms,
-9C₂=0
C₂=0

2C₂-9C₀=4
Therefore, C₀=-(4/9)

I am not sure about if I did it correctly or in the correct way. And I don't know what to do next.

 

[Deleted member 4993]

Hi all,

Consider a homogeneous linear ODE with constant coefficient: y''-9y=4+5sinh(3x), and the homogeneous solution is given by y_h (x)= C₁e^3x+C₂e^-3x.
But I don't know how to find out the particular solution, and the answer is y_p(x)=-(4/9)+(5/6)x cosh(3x).

Sorry for my poor English and thanks for your help!

Rewrite your homogeneous solution as:

y_H = A sinh(3x) + B cosh(3x)

That might help you think about the nature of the particular solution. We encounter this condition for "resonance without dampening" vibration problems.