Chapter 9 - Systems of Differential Equations - 9.1 First-Order Linear Systems - Problems - Page 588: 15

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Given $\frac{dx}{dt}-ty=\cos t\\ \frac{d^2y}{dt^2}-\frac{dx}{dt}+x=e^t$ We introduce new variables: $x_1=x\\ x_2=\frac{dx}{dt}\\ x_3=y\\ x_4=\frac{dy}{dt}$ Then the given differential equation can be replaced by $\frac{dx_1}{dt}-x_3t=\cos t\\ \frac{dx_4}{dt}-x_2+x_1=e^t$ Hence, $x_2=\frac{dx_1}{dt}\\ \frac{dx_1}{dt}=x_3t+\cos t\\ \frac{dx_4}{dt}=-x_1+x_2+e^t\\ x_4=\frac{dx_3}{dt}$