AboutScott A Wilson Expertise Most ordinary differential equations.
Experience I have taken Differential Equations at OSU.
Occasionally I have assisted people with questions on the subject.
Publications In the paper where MS students publish a thesis on shock waves and rarefaction fans.
Education/Credentials MS at OSU in Mathemematics.
MS at OSU in Mathematical Science.
Awards and Honors Graduation with honors for my BS and MS.
Past/Present Clients I answered many questions at OSU, down in south seattle at a college, at a church in Corvallis, OR,
as Safeway in Washington, and many other areas. I have had several thousand right here.
Question Solve the differential equation
y" - 8y' + 18y = 4e^(4x) + 3x^3 where the initial conditions are y(0)=2 and y'(0)=6
Answer The equation is y" - 8y' + 18y = 4e^(4x) + 3x^3.
The simplest solution that can be found would be y = Ae^(4x) + Bx³ + Cx² + Dx + C.
This leads to y' = 4Ae^(4x) + 3Bx² + 2Cx + D.
We then get y" = 16Ae^(4x) + 6Bx + 2C.
This will give rise to 5 equations with 5 unknown.
The first equation involves e^(4x).
It can be seen, since we have y" - 8y' + 18y, the equation should be
16Ae^(4x) - 8(4Ae^(4x)) + 18(Ae^(4x)) = 4e^(4x).
Dividing the entire equation by e^(4x) gives 16A - 32A + 18A = 4.
Now 16 + 18 = 34, and 34 - 32 = 2, so we have 2A = 4, so we can divide by 2 and see
that A = 2.
Since we only have one term that involves x³ on the left and 3x³ on the right,
we know that B = 3.
For the x², we have 24Bx² + 18Cx² = 0, and since B = 3, that gives 72 + 18C = 0.
This means that C = 4.
For x, we have 6Bx - 16Cx + 18Dx = 0, so putting in B and C, and dividing out the x, we get
6(3) - 16(4) + 18D = 0. That is, 18 - 64 + 18D = 0, so 18D = 46,
so D = 46/18 = 23/9.
For the constants, we have 2C - 8D + 18E = 0.
Putting in what C and D are, we get 2(4) - 8(23/9) + 18E = 0.
That simplifies to 18E = 112/9, or 9E = 56,
or E = 56/9.
Our equation is then y(x) = 2e^(4x) + 3x³ + 4x² + 23x/9 + 56/81.
This can be checked against the equation given and you will see that it is correct.
Now as far as the y(0) = 2 and the y'(0) = 6, you might have to add another term for each.
That is, start with y = Ae^(4x) + Bx^5 + Cx^4 + Dx³ + Ex² + Fx + G,
so y' = 4Ae^(4x) + 5Bx^4 + 4Cx³ + 3Dx² + 2Ex + F, and
y" = 16Ae^(4x) + 20Bx³ + 12Cx² + 6Dx + 2E.
Solve for all of the variables by setting up the same equations,
but now you will also need one for x^4 and one for x^5.
This will be two extra equations, and we can use y(0) = 2 and y'(0) = 6.
y(0) = 2 will generate A + G = 2.
y'(0) = 6 will generate 4 + F = 6.
