The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find
a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation. y'' − 7y' + 6y = x; y1 = ex
1 answer:
Answer:
y2 = (6x + 7)/36 + (Dx + E)e^x
Step-by-step explanation:
The method of reduction of order is applicable for second-order differential equations.
For a known solution y1 of a 2nd order differential equation, this method assumes a second solution in the form Uy1 which satisfies the said differential equation. It then assumes a reduced order for U'' (w' = U'').
The differential equation becomes easy to solve, and all that is left are integration and substitutions.
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The Laplace transform is then
Let denote the integral we want to compute. Integrating by parts, setting
gives
Integrate by parts again, setting
Then
3.122
The process to getting this comes in many steps. Firstly, you need to find the angles for JLK and MLJ. To find JLK use the arcsin function using the opposite side and the hypotenuse.
Arcsin(Opp/Hype) = JLK
Arcsin(.5) = JLK
30 degrees = JLK
This means MLJ = 31 degrees since they add up to 61 degrees.
Now we need to find the length of LJ, which we can do using the Pythagorean Theorem.
3^2 + JL^2 = 6^2
9 + JL^2 = 36
JL^2 = 27
JL =
Now that we have the angle of MLJ and the length of JL, we can use the tangent function to find MJ.
Tan(angle) = opp/adj
Tan(31) = MJ/
Tan(31) = MJ
3.122 = MJ
The process to getting this comes in many steps. Firstly, you need to find the angles for JLK and MLJ. To find JLK use the arcsin function using the opposite side and the hypotenuse.
Arcsin(Opp/Hype) = JLK
Arcsin(.5) = JLK
30 degrees = JLK
This means MLJ = 31 degrees since they add up to 61 degrees.
Now we need to find the length of LJ, which we can do using the Pythagorean Theorem.
3^2 + JL^2 = 6^2
9 + JL^2 = 36
JL^2 = 27
JL =
Now that we have the angle of MLJ and the length of JL, we can use the tangent function to find MJ.
Tan(angle) = opp/adj
Tan(31) = MJ/
3.122 = MJ