Differential Equations and Linear Algebra, 4th Edition

Section 7.1

1. 1. $x_1^{\text{′}}=x_2\text{,}{x}_2^{\text{′}}=-7x_1-3x_2+t^2$

2. 2. $x_1^{\text{′}}=x_2\text{,}{x}_2^{\text{′}}=x_3\text{,}{x}_3^{\text{′}}=x_4,$$x_4^{\text{′}}=-x_1+3x_2-6x_3+\cos 3t$

3. 3. $x_1^{\text{′}}=x_2\text{,}{t}^2{x}_2^{\text{′}}=(1-t^2)x_1-t{x}_2$

4. 4. $x_1^{\text{′}}=x_2\text{,}{x}_2^{\text{′}}=x_3\text{,}{t}^3{x}_3^{\text{′}}=-5x_1-3{tx}_2+2t^2{x}_3+\ln t$

5. 5. $x_1^{\text{′}}=x_2\text{,}{x}_2^{\text{′}}=x_3\text{,}{x}_3^{\text{′}}=x_2^2+\cos x_1$

6. 6. $x_1^{\text{′}}=x_2\text{,}{x}_2^{\text{′}}=5x_1-4y_1\text{,}{y}_1^{\text{′}}=y_2\text{,}{y}_2^{\text{′}}=-4x_1+5y_1$

7. 7. $x_1^{\text{′}}=x_2\text{,}{y}_1^{\text{′}}=y_2\text{,}{x}_2^{\text{′}}=-k{x}_1·{(x_1^2+y_1^2)}^{-3/2}\text{,}{y}_2^{\text{′}}=-k{y}_1·{(x_1^2+y_1^2)}^{-3/2}$

8. 8. $x_1^{\text{′}}=x_2\text{,}{x}_2^{\text{′}}=-4x_1+2y_1-3x_2$ $y_1^{\text{′}}=y_2\text{,}{y}_2^{\text{′}}=3x_1-y_1-2y_2+\cos t$

9. 9. $x_1^{\text{′}}=x_2$, $y_1^{\text{′}}=y_2\text{,}{z}_1^{\text{′}}=z_2\text{,}{x}_2^{\text{′}}=3x_1-y_1+2z_1\text{,}{y}_2^{\text{′}}=x_1+y_1-4z_1\text{,}{z}_2^{\text{′}}=5x_1-y_1-z_1$

10. 10. $x_1^{\text{′}}=x_2\text{,}{x}_2^{\text{′}}=x_1(1-y_1)$ $y_1^{\text{′}}=y_2\text{,}{y}_2^{\text{′}}=y_1(1-x_1)$

11. 11. $x(t)=A\cos t+B\sin t\text{,}y(t)=B\cos t-A\sin t$

    

12. 12. $x(t)=A{e}^t+B{e}^{-t}\text{,}y(t)=A{e}^t-B{e}^{-t}$

13. 13. $x(t)=A\cos 2t+B\sin 2t\text{,}y(t)=-B\cos 2t+A\sin 2t$; $x(t)=\cos 2t\text{,}y(t)=\sin 2t$