Homework Four

35

$$y(x) = C_1 e^{-2 x} + C_2 e^x -x - \frac{1}{2},$$

$$y(x) = e^{-2 x} -\frac{1}{2} e^x -x - \frac{1}{2}.$$

36

$$y(x) = C_1 \cos 2 x+ C_2 \sin 2 x + \frac{x^2}{4} - \frac{1}{8} + \frac{3}{5}e^x,$$

$$y(x)=-frac{19}{40}\cos 2 x+\frac{7}{10}\sin 2 x+\frac{x^2}{4}-\frac{1}{8}+\frac{3}{5}e^x.$$

37

$$y(x) = C_1 e^x +c_2 x e^x+ \frac{1}{6}x^3 e^x + 4, C_1 = -3, C_2 = 4,$$

$$y(x) = -3 e^x +4 x e^x+ \frac{1}{6}x^3 e^x + 4.$$

38

$$y_1 (x) = x( A x^4 + B x^3 + C x^2 _ D x +E),$$

$$y_2(x) = x( F x^2 + G x + H) e^{-3 x},$$

$$y_3(x) = L \sin 3 x + M \cos 3 x.$$

39

$$y_1(x) = A e^{-x},$$

$$y_2(x) =x\left[ ( B x^2 + C x + D) e{-x} \cos{x} + (E x^2 + F x + G) e^{-x}\sin x \right].$$

40

$$y_1(x) = A x^2 + B x + C,$$

$$y_2(x)= x^2 (D x + E) e^{2 x},$$

$$y_3(x) = ( F x + G) \sin 2 x + (H x + I) \cos{2 x}.$$

41

$$y(x) = C_1 + C_2 e^{- 2 x} + \frac{3}{2} = \frac{1}{2}(\sin 2 x+ \cos 2 x).$$

42

$$y(x) = C_1 e^{-\frac{x}{2}} + C_2 e^{-x} + x^2 - 6 x +14-\frac{3}{10} \sin x - \frac{9}{10}\cos x.$$

43

$$u(t) = C_1 \cos \omega_0 t + C_2 \sin \omega_0 t + \frac{1}{2 \omega_0} t \sin \omega_0 t.$$

44

$$y_1(x) = A x + B,$$

$$y_2(x) = x \left[ ( C x + D ) \sin x + (E x + F ) \cos x \right].$$

45

$$y_1(x) = A e^x \cos 2 x + B e^x \sin 2 x,$$

$$y_2(x) = (C x + D ) e^{2 x} \sin x + (E x + F) e^{2 x } \cos x.$$