User:Egm6321.f12.team4.lu/Homework6 R6.5&R6.9&R6.6(2.7,2.8,2.9)

Problem R*6.5-Particular solution by using variation parameters method
sec11-3 sec11-4 sec11-5 sec12-2 sec32-2

Statement
Recall p.12-2 R*3.3 Use the same idea of variation of constants(parameters) to find the particular solution $$y_P(x)$$ after knowing the homogeneous solution $$y_H(x)$$, i.e. consider the following solution $$y(x)=A(x)y_H(x)$$, with $$A(x)$$ being the unknown to be found.

Solution
Recall

Derive Eq(6.5.1) we obtain

Derive Eq(6.5.3) we obtain

Substitute Eq(6.5.5) into Eq(6.5.4)

Substitute Eq(6.5.1) and Eq(6.5.6) into Eq(6.5.2), we obtain

Rearrange Eq(6.5.7)

Integrate Eq(6.5.8)

Finally, we obtain the particular solution is

where

End

Author and References

 * Solved and Typed by -- Jinchao Lu
 * Reviewed by --

Problem R*6.9-2nd homogeneous solution by variation of parameters
sec34-4 sec34-5 sec35-3 sec35-4

Statement
For the L2-ODE-VC(1) p.35-3,

select a valid homogeneous solution, and call it $$u_1$$.

Find the 2nd homogeneous solution $$u_2(x)$$ by variation of parameters, and compare to $$e^{xr_2(x)}$$

Solution
Trial solution:$$y=e^{rx},r=constant$$

Characteristic equation:

WA:

From R*6.8, we know that $$r_2(x)$$ is not a valid root. Therefore,

Recall,

Determine $$a_1(x)$$

Hence,

Substitute Eq(6.9.2) and Eq(6.9.5) into Eq(6.9.3), we obtain Hence,

Substitute Eq(6.9.2) and Eq(6.9.6) into Eq(6.9.4), we obtain

Finally, we obtain

Compare $$ u_2(x)=-x $$ to $$u_2(x)=e^{xr_2(x)}$$, we can know that $$u_2(x)=e^{xr_2(x)}$$ is not a valid solution.

End

Author and References

 * Solved and Typed by -- Jinchao Lu
 * Reviewed by --

Problem R*6.6-Solve Nonhomogeneous L2-ODE-CC
sec32-5 sec33-5 sec33-6

Statement
2.7. Verify result with table of particular solutions for $$f(t)=texp(bt)$$

2.8. Solve the nonhomogeneous L2-ODE-CC (1)sec32-5

with the following excitation:

Gaussian distribution:

$$f(t)=exp(-t^2)$$

For the coefficients $$(a_0,a_1,a_2)$$, consider two different characteristic equations:

2.8.1   $$(r+1)(r-2)=0$$

2.8.2   $$(r-4)^2=0$$

2.9. For each case in 2.8.1 and 2.8.2, determine the fundamental period of undamped free vibration. Plot the homogeneous soln $$y_H(t)$$ for about 5 periods, the particular solution $$y_P(t)$$ for the excitation (3)-(4) p33-5 for the same time interval, and the complete solution $$y(t)$$, assuming zero initial conditions.

2.7
Recall

And

Substitute Eq(2.7.2) into Eq(2.7.1), we obtain

Finally,we obtain

where

Compare to the table

They match.

End

2.8.1
Recall

And

We obtain

Substitute Eq(2.8.1.2) and Eq(2.8.1.3) into Eq(2.8.1.1), we obtain

NOTE: Wolfram|Alpha was used to determine the following integrals:

integrate e^(-t-t^2)dt

http://www.wolframalpha.com/input/?i=integrate+e^%28-t-t^2%29dt

integrate e^(3s)1/2 e^(1/4) sqrt(π) erf(1/2+s)ds

http://www.wolframalpha.com/input/?i=integrate+e^%283s%291%2F2+e^%281%2F4%29+sqrt%28%CF%80%29+erf%281%2F2%2Bs%29ds

2.8.2
Similar to 2.8.1 above

we have

Recall

And

Substitute Eq(2.8.2.1) and Eq(2.8.2.3) into Eq(2.8.2.2), we obtain

NOTE: Wolfram|Alpha was used to determine the following integrals:

integrate e^(4t-t^2)dt

http://www.wolframalpha.com/input/?i=integrate+e^%284t-t^2%29dt

integrate -1/2 e^4 sqrt(π) erf(2-t) dt

http://www.wolframalpha.com/input/?i=integrate+-1%2F2+e^4+sqrt%28%CF%80%29+erf%282-t%29+dt

Author and References

 * Solved and Typed by -- Jinchao Lu
 * Reviewed by --