Question

The general solution of the nonhomogeneous differential equation $y^{\prime \prime }+\alpha y^{\prime }+\beta y=g(t)$ is given, where $c_1$ and $c_2$ are arbitrary constants. Determine the constants $\alpha$ and $\beta$ and the function $g(t)$. $$y(t)=c_1{e}^t+c_2{e}^{2t}+2e^{-2t}$$

Short answer

In summary, the given nonhomogeneous differential equation is: $$y^{″}(t)-6y^{\prime }(t)+12y(t)=8$$ with the general solution: $$y(t)=c_1{e}^t+c_2{e}^{2t}+2e^{-2t}$$

Step-by-step solution

Find the first and second derivatives of y(t)

To find the first derivative y'(t), we need to differentiate y(t) with respect to t: $$y^{\prime }(t)=\frac{d}{dt}(c_1{e}^t+c_2{e}^{2t}+2e^{-2t})$$ By applying the standard rules for derivatives, we obtain: $$y^{\prime }(t)=c_1{e}^t+2c_2{e}^{2t}-4e^{-2t}$$ Next, we'll find the second derivative y''(t) by differentiating y'(t) with respect to t: $$y^{″}(t)=\frac{d}{dt}(c_1{e}^t+2c_2{e}^{2t}-4e^{-2t})$$ By applying the standard rules for derivatives, we obtain: $$y^{″}(t)=c_1{e}^t+4c_2{e}^{2t}+8e^{-2t}$$

Substitute y(t), y'(t), and y''(t) into the given differential equation

We can now substitute the expressions for y(t), y'(t), and y''(t) into the given differential equation: $$y^{″}(t)+\alpha y^{\prime }(t)+\beta y(t)=g(t)$$ Substituting, we get: $$(c_1{e}^t+4c_2{e}^{2t}+8e^{-2t})+\alpha (c_1{e}^t+2c_2{e}^{2t}-4e^{-2t})+\beta (c_1{e}^t+c_2{e}^{2t}+2e^{-2t})=g(t)$$

Collect terms and find α, β, and g(t)

By combining terms with the same exponential factor, we arrive at a new system of equations: $$(1+\alpha +\beta )c_1{e}^t+(2\alpha +\beta )c_2{e}^{2t}+(\alpha -4)e^{-2t}+8e^{-2t}=g(t)$$ To satisfy this equation, we can match the coefficients of each exponential term on both sides of the equation. By doing this, we can find the values for α, β, and g(t) as follows: For the term $e^t$: $$1+\alpha +\beta =0$$ For the term $e^{2t}$: $$2\alpha +\beta =0$$ For the term $e^{-2t}$: $$\alpha -4=g(t)$$ $$8=g(t)$$ Solving the system of equations, we get: $$\alpha =-6$$ $$\beta =12$$ $$g(t)=8$$ Therefore, the constants α and β are -6 and 12, respectively, and the function g(t) is 8.