Obtain the differential equation from the following:
1. y=C(subscript 1)e^x+ C(subscript 2)xe^x
2. y=[C(subscript 1)e^2x]cos3x+[C(subscript 2)e^2x]sin3x
Denoting the diferential operator d/dx by D, we can write:
D exp(a x) = a exp(a x) ------->
(D - a) exp(x) = 0
If we have a term x exp(x), then applying D-1 gives:
(D - 1) x exp(x) = exp(x)
Applying D - 1 again will annihilate the remaining exp(x):
(D-1)^2 x exp(x) = (D-1) exp(x) = 0
So, the differential operator in problem 1) is (D-1)^2 = D^2 - 2 D + 1
In problem 2), you can write the function in terms of exp[(2+3i) x] and exp[(2-3i)x]. The operator that annihilates the term exp[(2+3i) x] is
O1 = D - (2+3i)
The operator that annihilates the term exp[(2-3i) x] is
O2 = D - (2-3i)
The perator that will annihikate both terms can then be taken to be O1O2. Obviously if O1 f = 0, then O1O2f = O2O1f = 0.
We have with z = 2+3i
O1O2 = (D - z ) (D - z*) =
D^2 - 2 Re(z) D + |z|^2 =
D^2 - 4 D + 13
To obtain the differential equation from the given expressions, we need to differentiate the functions with respect to the independent variable.
1. For y = C₁e^x + C₂xe^x:
To find the differential equation, we differentiate y with respect to x:
dy/dx = d/dx (C₁e^x + C₂xe^x)
Now let's differentiate each term separately:
The derivative of C₁e^x with respect to x is C₁e^x, since e^x is the derivative of itself.
The derivative of C₂xe^x with respect to x requires the product rule. Applying the product rule, we get:
d/dx (C₂xe^x) = C₂(d/dx(x)e^x) + (x)(d/dx(e^x))
= C₂(e^x + xe^x) + (x)(e^x)
= C₂xe^x + C₂e^x + xe^x
Now adding up the derivatives:
dy/dx = C₁e^x + C₂xe^x + C₂e^x + xe^x
dy/dx = (C₁ + C₂)e^x + (C₂ + 1)xe^x
Therefore, the differential equation is:
dy/dx = (C₁ + C₂)e^x + (C₂ + 1)xe^x
2. For y = [C₁e^(2x)]cos(3x) + [C₂e^(2x)]sin(3x):
To find the differential equation, we differentiate y with respect to x:
dy/dx = d/dx ([C₁e^(2x)]cos(3x) + [C₂e^(2x)]sin(3x))
Now let's differentiate each term separately:
Using the product rule, we have:
d/dx (C₁e^(2x) cos(3x)) = C₁(d/dx(e^(2x))cos(3x)) + e^(2x)(d/dx(cos(3x)))
= C₁(2e^(2x)cos(3x)) + e^(2x)(-3sin(3x))
Similarly,
d/dx (C₂e^(2x)sin(3x)) = C₂(2e^(2x)sin(3x)) + e^(2x)(3cos(3x))
Now adding up the derivatives:
dy/dx = C₁(2e^(2x)cos(3x)) + e^(2x)(-3sin(3x)) + C₂(2e^(2x)sin(3x)) + e^(2x)(3cos(3x))
dy/dx = (2C₁e^(2x)cos(3x) + 2C₂e^(2x)sin(3x)) + (e^(2x)(-3sin(3x) + 3cos(3x)))
Therefore, the differential equation is:
dy/dx = (2C₁e^(2x)cos(3x) + 2C₂e^(2x)sin(3x)) + (e^(2x)(-3sin(3x) + 3cos(3x)))