Show all steps and give appropriate justifications. Simplify your answers whenever possible. 1. Solve the IVP y(3) – y" = 6x + 2e", y(0) = 0, y'(0) = 4, y"(0) : 2. Find a particular solution of y" + 16y = sin(4.x)+ sec²(4x)
![Show all steps and give appropriate justifications. Simplify your answers whenever possible.
1. Solve the IVP
y(3) – y" = 6x + 2e", y(0) = 0, y'(0) = 4, y"(0) :
2. Find a particular solution of
y" + 16y = sin(4.x)+ sec²(4x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F070ceba2-dab6-437b-be0c-994289d3c6b2%2Ffa0d756c-76ac-452a-9fb8-9ddc55c49cb6%2F6qj44l8.jpeg&w=3840&q=75)
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Here, we are given the IVP,
.
Note:
So, first, we will find the solution to the corresponding homogeneous equation.
The characteristic equation is,
m3 - m2 = 0
=> m2(m-1) = 0
=> m = 0, 0, 1.
Thus, yc = c1 + c2x + c3ex.
Let us solve this differential equation by the method of undetermined coefficients.
Here, g(x) = 6x + 2ex.
For 6x, the possible choice of solution is A+Bx. But x is already solving the corresponding homogeneous solution.
So, now we multiply by x to the choice.
Then, for 6x, the possible choice of solution is Ax+Bx2. But again x is already solving the corresponding homogeneous solution.
So, now we multiply by x to the choice.
Then finally, for 6x, the possible choice of solution is Ax2+Bx3.
Now, the possible choice of solution is Cex. But ex is already solving the corresponding homogeneous solution.
So, now we multiply by x to the choice.
Then finally, for ex, the possible choice of solution is Cxex.
Thus, yp = Ax2+Bx3+Cxex.
Now, we need to find the constants A, B, and C.
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