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Linear Differential Equation: (d^2-3d+2)y=sin(e^-x)
In this article, we will discuss the solution of the linear differential equation (d^2-3d+2)y=sin(e^-x). This equation is a second-order linear ordinary differential equation with a trigonometric function on the right-hand side.
Standard Form
The given equation can be written in the standard form of a linear differential equation as:
$y'' - 3y' + 2y = \sin(e^{-x})$
where $y''$ is the second derivative of $y$ with respect to $x$, and $y'$ is the first derivative of $y$ with respect to $x$.
Complementary Function
To find the complementary function, we need to solve the homogeneous equation:
$y'' - 3y' + 2y = 0$
The characteristic equation of this homogeneous equation is:
$r^2 - 3r + 2 = 0$
Solving the characteristic equation, we get:
$(r - 1)(r - 2) = 0$
Thus, the roots of the characteristic equation are $r_1 = 1$ and $r_2 = 2$. The complementary function is:
$y_c(x) = c_1e^x + c_2e^{2x}$
Particular Solution
To find the particular solution, we use the method of undetermined coefficients. Since the right-hand side of the equation is $\sin(e^{-x})$, we assume a particular solution of the form:
$y_p(x) = A\sin(e^{-x}) + B\cos(e^{-x})$
Substituting this into the original equation, we get:
$-Ae^{-x}\sin(e^{-x}) - Ae^{-x}\cos(e^{-x}) - 3Ae^{-x}\sin(e^{-x}) + 3Ae^{-x}\cos(e^{-x}) + 2A\sin(e^{-x}) + 2B\cos(e^{-x}) = \sin(e^{-x})$
Equating the coefficients of $\sin(e^{-x})$ and $\cos(e^{-x})$, we get:
$-A - 3A + 2A = 1$ $-A + 3A + 2B = 0$
Solving these equations, we get:
$A = -\frac{1}{4}$ $B = -\frac{1}{4}$
Thus, the particular solution is:
$y_p(x) = -\frac{1}{4}\sin(e^{-x}) - \frac{1}{4}\cos(e^{-x})$
General Solution
The general solution of the differential equation is the sum of the complementary function and the particular solution:
$y(x) = c_1e^x + c_2e^{2x} - \frac{1}{4}\sin(e^{-x}) - \frac{1}{4}\cos(e^{-x})$
This is the final solution of the linear differential equation (d^2-3d+2)y=sin(e^-x).
Conclusion
In this article, we have solved the linear differential equation (d^2-3d+2)y=sin(e^-x) using the method of undetermined coefficients. The general solution of the equation is a combination of the complementary function and the particular solution. The particular solution is obtained by assuming a solution of the form $A\sin(e^{-x}) + B\cos(e^{-x})$ and equating the coefficients of $\sin(e^{-x})$ and $\cos(e^{-x})$.
