Question

Use the method of variation of parameters to determine the general solution of the given differential equation. $$ y^{\prime \prime \prime}-2 y^{\prime \prime}-y^{\prime}+2 y=e^{4 t} $$

Short answer

Question: Determine the general solution for the given third-order linear differential equation using the method of variation of parameters: $$y^{\prime \prime \prime}-2 y^{\prime \prime}-y^{\prime}+2 y=e^{4t}$$ Answer: The general solution for the given third-order linear differential equation using the method of variation of parameters is: $$y(t) = C_1e^{-t} + C_2e^{t} + C_3e^{2t} + (-5t)e^{-t} - te^{t}$$, where \(C_1\), \(C_2\), and \(C_3\) are arbitrary constants.

Step-by-step solution

Characteristic Equation of the Homogeneous Equation

First, let's find the homogeneous equation related to the given differential equation: $$ y^{\prime \prime \prime}-2 y^{\prime \prime}-y^{\prime}+2 y=0 $$ To find the characteristic equation, we will replace each derivative with an exponent. Let's assume a solution of the form \(y=e^{rt}\) for the homogeneous equation. For the given equation, the characteristic equation becomes: $$ r^3 - 2r^2 - r + 2 = 0 $$

Finding the Roots of the Characteristic Equation

The characteristic equation is a cubic equation. We should try to find at least one real root which will help factorize the equation into smaller polynomials to find the remaining roots. By trial and error or synthetic division, one will observe that \(r=1\) is a root. Now we can factor the equation as: $$ (r-1)(r^2-r-2)=0 $$ The quadratic equation has roots: $$ r_{1,2}=\frac{1\pm \sqrt{1^2-4(-2)}}{2}=\frac{1\pm \sqrt{9}}{2}=\frac{1\pm 3}{2} $$ Thus, the roots are \(r_1=-1\), \(r_2=1\), and \(r_3=2\).

Finding the Complementary Function

Using the roots \(-1\), \(1\), and \(2\), we can write the complementary function for the given homogeneous equation as: $$ y_c = C_1e^{-t} + C_2e^{t} + C_3e^{2t} $$ where \(C_1, C_2\), and \(C_3\) are constants.

Applying Variation of Parameters

Next, let's use the method of variation of parameters to find the particular solution. We need to take the first derivative of the complementary solution: $$ y_c'= -C_1e^{-t} + C_2e^{t} + 2C_3e^{2t} $$ To find the particular solution, we will assume it in the form: $$ y_p =u_1e^{-t}+u_2e^{t}+u_3e^{2t} $$ Now, we need to differentiate \(y_p\): $$ y_p' = -u_1'e^{-t}+u_1e^{-t}+u_2'e^{t}+u_2e^{t}+ 2u_3'e^{2t} +2u_3e^{2t} $$ We will differentiate \(y_p'\) once more to get: $$ y_p'' = u_1'e^{-t}+2u_1e^{-t}+u_2'e^{t}+2u_2e^{t}+ 4u_3'e^{2t} +4u_3e^{2t} $$ Now we take the third derivative of \(y_p\): $$ y_p''' = -u_1'e^{-t}+3u_1e^{-t}+u_2'e^{t}+3u_2e^{t}+ 8u_3'e^{2t} +8u_3e^{2t} $$

Determining the Particular Solution

Now let's substitute \(y_p'\)', \(y_p''\), and \(y_p'''\) into the given differential equation: $$ y_p''' - 2y_p'' - y_p' + 2y_p = e^{4t} $$ Substitute: $$ (-u_1'e^{-t}+3u_1e^{-t}+u_2'e^{t}+3u_2e^{t}+ 8u_3'e^{2t} +8u_3e^{2t})-2(u_1'e^{-t}+2u_1e^{-t}+u_2'e^{t}+2u_2e^{t}+ 4u_3'e^{2t} +4u_3e^{2t}) - (-u_1'e^{-t}+u_1e^{-t}+u_2'e^{t}+u_2e^{t}+ 2u_3'e^{2t} +2u_3e^{2t}) +2(u_1e^{-t}+u_2e^{t}+u_3e^{2t}) = e^{4t} $$ Simplify and collect terms: $$ (-u_1'e^{-t}+5u_1e^{-t} - u_2'e^{t}+u_2e^{t}+ 6u_3'e^{2t} +10u_3e^{2t}) = e^{4t} $$ Now we compare the coefficients of the exponential of like powers: $$ -u_1'=5, -u_2'=1, 6u_3'=0 $$ Integrating each equation with respect to t yields: $$ u_1(t) = -5t + C_1, u_2(t) = -t + C_2, u_3(t) = C_3 $$ Now, plugging \(u_1(t)\), \(u_2(t)\), and \(u_3(t)\) back into the expression for \(y_p\): $$ y_p =(-5t + C_1)e^{-t} + (-t + C_2)e^{t} + C_3e^{2t} $$ Since \(y_p\) is a particular solution and we want the first function linearly independent of the complementary function, we can take \(C_1 = C_2 = C_3 = 0\). This gives: $$ y_p = (-5t)e^{-t} - te^{t} $$

General Solution

Now, the general solution is the sum of the complementary and particular solutions: $$ y(t) = y_c + y_p = C_1e^{-t} + C_2e^{t} + C_3e^{2t} + (-5t)e^{-t} - te^{t} $$ This is the general solution of the given third-order differential equation using the method of variation of parameters.