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Linear Differential Equations: Solving (d^2+1)y=sinx
In this article, we will explore the solution to the linear differential equation (d^2+1)y=sinx. This equation is a classic example of a second-order linear ordinary differential equation (ODE) with a sinusoidal forcing term.
The Equation
The equation is given by:
(d^2+1)y = sinx
where y is the dependent variable, x is the independent variable, and d is the differential operator.
Homogeneous Solution
To solve this equation, we first need to find the homogeneous solution, which is the solution to the equation when the right-hand side is zero. In this case, the homogeneous equation is:
(d^2+1)y = 0
To solve this equation, we can use the characteristic equation, which is:
r^2 + 1 = 0
This equation has two complex roots:
r = ±i
Using these roots, we can write the homogeneous solution as:
y_h = c1cosx + c2sinx
where c1 and c2 are constants.
Particular Solution
To find the particular solution, we need to find a function y_p that satisfies the original equation:
(d^2+1)y_p = sinx
Using the method of undetermined coefficients, we assume that the particular solution has the form:
y_p = Acosx + Bsinx
Substituting this into the original equation, we get:
(-Acosx - Bsinx) + (Acosx + Bsinx) = sinx
Simplifying this equation, we get:
-Acosx + Bsinx = sinx
Equating the coefficients of cosx and sinx, we get:
A = 0 B = 1/2
Therefore, the particular solution is:
y_p = (1/2)sinx
General Solution
The general solution to the equation is the sum of the homogeneous solution and the particular solution:
y = y_h + y_p = c1cosx + c2sinx + (1/2)sinx
Simplifying this equation, we get:
y = c1cosx + (c2 + 1/2)sinx
This is the general solution to the linear differential equation (d^2+1)y=sinx.
Conclusion
In this article, we have solved the linear differential equation (d^2+1)y=sinx using the method of undetermined coefficients. The general solution to the equation is a linear combination of cosx and sinx. This equation is an important example of a second-order linear ODE with a sinusoidal forcing term, and its solution has many applications in physics, engineering, and other fields.
