y%3D0.jpeg "(d^4+2d^3+10d^2)y=0")
Solving the Differential Equation: (d^4+2d^3+10d^2)y=0
In this article, we will solve the differential equation:
(d^4+2d^3+10d^2)y=0
where y is a function of x, and d denotes the differential operator with respect to x.
Step 1: Factorization
The first step in solving this differential equation is to factorize the left-hand side:
(d^4+2d^3+10d^2)y = d^2(d^2+2d+10)y = 0
Step 2: Solving for y
Now, we can see that the differential equation can be written as a product of two factors:
d^2y = 0 or (d^2+2d+10)y = 0
Case 1: d^2y = 0
The general solution to the equation d^2y = 0 is:
y = Ax + B
where A and B are arbitrary constants.
Case 2: (d^2+2d+10)y = 0
To solve this equation, we can use the characteristic equation:
r^2 + 2r + 10 = 0
Solving for r, we get:
r = -1 ± 3i
Thus, the general solution to the equation (d^2+2d+10)y = 0 is:
y = e^(-x)(Ccos(3x) + Dsin(3x))
where C and D are arbitrary constants.
General Solution
Combining the solutions from Case 1 and Case 2, we get the general solution to the original differential equation:
(d^4+2d^3+10d^2)y = 0
y = Ax + B + e^(-x)(Ccos(3x) + Dsin(3x))
where A, B, C, and D are arbitrary constants.
This is the general solution to the given differential equation.
