1. Differential Equations
Higher-Order D.E. with constant coefficients
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2. The general solution will always have the form:
Here is the general form of an n-th order linear equation with constant coefficients.
The general solution will always have the form:
yh is the solution to the corresponding homogeneous equation, where g(t)=0
yp is a particular solution to the original DE.
To find the homogeneous solution we can find the roots of the characteristic equation, just like we did for 2nd-order equations.
The particular solution can be found by undetermined coefficients if the g(t) function has a convenient form, or a generalized version of variation of parameters.
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3. Here are a few examples:
The solution depends on being able to find the roots of the characteristic equation. If it factors nicely this is no problem, but can be tricky for higher orders.
Here are a few examples:
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5. The characteristic equation for this one is:
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6. The characteristic equation for this one is:
The trick is to recognize that this only has even powers, so we can use the quadratic formula, thinking of r2 as our variable. This one factors out nicely:
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7. The characteristic equation for this one is:
The trick is to recognize that this only has even powers, so we can use the quadratic formula, thinking of r2 as our variable. This one factors out nicely:
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8. The characteristic equation for this one is:
The trick is to recognize that this only has even powers, so we can use the quadratic formula, thinking of r2 as our variable. This one factors out nicely:
There are 4 real roots, so the solution is:
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9. Here is the characteristic equation:
r=0 is definitely a solution, as is r=2.
But there should be 4 solutions! How do you get the other two?
They are actually complex numbers! We have a couple of options from here.
Option 1: Long division
Option 2: Convert to Polar form
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10. Since r=2 solves the equation, (r-2) must be a factor of (r3-8).
Option 1: Long Division
Since r=2 solves the equation, (r-2) must be a factor of (r3-8).
Divide out the factor (r-2) to find the remaining quadratic expression.
Now use the quadratic formula on the remaining expression.
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11. Option 2: Convert to Polar form
Any complex number can be written in polar form with these formulas:
In our problem we want to write the number 8 in polar form.
imaginary
2∏/3
real
4∏/3
The equation we want to solve is:
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12. Here is the characteristic equation:
There are 4 roots:
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14. First solve the homogeneous equation:
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15. First solve the homogeneous equation:
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16. First solve the homogeneous equation:
Next make a guess for the particular solution:
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17. First solve the homogeneous equation:
Next make a guess for the particular solution:
Comparing to yh notice that the Aet term is a solution to the homogeneous problem. So we have to modify our guess.
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18. The homogeneous solution is
The form for the particular solution is
Find the derivatives of yp then substitute to find the constants A, B and C
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For Campus Learning Assistance Services at UCSB