Solving 4th Order Differential Equation: y^(4)-1=5

[ranger1716]

I have a question regarding a 4th order differential equation from an exam i just took.

we were asked to solve y^(4)-1=5 given y'(0)=y''(0)=y^(3)(0)=0

I started by factoring down to (r-1)(r+1)(r^2+1)=5.

I then found my general solution to be y=C_1e^6x+C_2e^4x+C_3e^2x+C_4e^-2x

Obviously I would then be left with four equations with four unknowns to solve for my constants. Would I need to use a solver and/or hand solve the equations in order to find the constants? I didn't have time to do that so I just put the equations into a matrix and said that the constants were all equal to zero.

Just thought I would ask what the right approach would be (a little to anxious to wait another week)

 

[Data]

Can you clarify what your DE is:

Is it y^{(4)} - y = 5, or y^{(4)} - 1 =5?

In the second case, rearrange it to y^{(4)} = 6. The answer is just some 4th order polynomial (the coeffs are easy so I'll let you work them out. Think about it for a minute ).

In the first case the answer is even simpler; y(x) = -5.

Edit: Ignore this and see below.

 

[AiRAVATA]

If what you wrote is correct, then the solution is y(x)=\frac{x^4}{4}+c, where c is a constant.

If what you meant is y^{(4)}-y=5, then you are missing one condition on y(0), and your solution is incorrect.

What went wrong you may ask. Well, the solution of the second equation is the sum of the homogeneous equation y_h^{(4)}-y_h=0, plus the solution of the particular equation y_p^{(4)}-y_p=5. The particular solution is clearly y_p=-5, so you only have to solve the homogeneous equation, which indeed has the form

y_h(x)=c_1e^{r_1x}+c_2e^{r_2x}+c_3e^{r_3x}+c_4e^{r_4x},

where r_i are the roots of the characteristic equation r^4-1=0. Therefore, the solution is

y_h(x)=c_1e^{x}+c_2e^{-x}+c_3e^{ix}+c_4e^{-ix},

and evaluating on the initial conditions,

y(x)=\frac{y_0+5}{2}\left[\cosh x+\cos x\right]-5,

where y(0)=y_0.

 

[Data]

Whoops, you're right. I forgot about the missing fourth init. cond.!

 

[AiRAVATA]

Lets hope the missing condition is y(0)=-5 ;)

 

[Data]

Indeed! I was about to post the problem with regard to the 2 in front of the cos term (I needed to check that I wasn't making another dumb mistake, first!), but I see that you've already gotten to it.

 

[AiRAVATA]

Lol, I always have to edit my posts like 7 times :P

 

[Data]

Yes, it seems like a common phenomenon with mine as well .