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  1. \documentclass[11pt,a4paper]{article}
  2. \usepackage{amsmath}
  3. \setlength{\parindent}{0mm}
  4. \begin{document}
  5. Lista 7 zadanie 39 \\
  6. Nastepujace uklady rownan rozwiazac metoda operatorowa lub sprowadzajac je do ukladow rownan rozniczkowych rzedu pierwszego w postaci normalnej. \\
  7. \begin{math}
  8. \\
  9. \begin{cases}
  10. x''+3x-y=t \\
  11. y''-9x-5y= \cos t
  12. \end{cases}
  13. \\ \\
  14. \begin{cases}
  15. D^2x+3x-y=t \\
  16. D^2y-9x-5y= \cos t
  17. \end{cases}
  18. \\ \\
  19. \begin{cases}
  20. \left(D^2+3 \right) x-y=t \\
  21. -9x+ \left(D^2-5 \right)y= \cos t
  22. \end{cases}
  23. \\ \\
  24. W= \left|
  25. \begin{array}{cc}
  26. D^2+3 & -1 \\
  27. -9 & D^2-5
  28. \end{array}
  29. \right| = \left(D^2+3 \right) \left(D^2-5 \right)-9=D^4-2D^2-24 \\
  30. \\ W_{x} = \left|
  31. \begin{array}{cc}
  32. t & -1 \\
  33. \cos t & D^2-5
  34. \end{array}
  35. \right| = \left(D^2-5 \right)t+ \cos t=D^2t-5t+ \cos t= \cos t -5t \\
  36. \\ W_{y} = \left|
  37. \begin{array}{cc}
  38. D^2+3 & t \\
  39. -9 & \cos t
  40. \end{array}
  41. \right| = \left(D^2+3 \right) \cos t+9t=D^2 \left( \cos t \right)+3 \cos t+9=
  42. \\ = - \cos t+3 \cos t +9t=2 \cos t+9t \\
  43. \\ (1) \hspace{10} \left(D^4-2D^2-24 \right)x= \cos t-5t -> x^{IV}-2x''-24x= \cost-5t \\
  44. \\ (2) \hspace{10} \left(D^4-2D^2-24 \right)y=2 \cos t+9t -> y^{IV}-2y''-24y=2 \cos t+9t \\
  45. \\ (1): \\
  46. \\ x^{IV}-2x''-24= \cos t-5t \\
  47. \\ l^4-2l^2-24=0 \\
  48. \\ l_1= -2i, \ l_2=2i, \ l_3= -\sqrt{6}, \ l_4= \sqrt{6} \\
  49. \\ x=C_1 \cos 2t+C_2 \sin 2t+C_3e^{- \sqrt{6}t}+C_4e^{\sqrt{6}t} \\
  50. \\ x^{IV}-2x''-24x= \cos t\\
  51. \\ A \cos t+2A \cos t-24A \cos t= \cos t\\
  52. \\ A= - \frac{1}{21} \\
  53. \\ x^{IV}-2x''-24x= -5t \\
  54. \\ 24 \left(Ct+D \right)= -5t \\
  55. \\ 24Ct+24D= -5t-> D=0, \ C= - \frac{5}{24} \\
  56. \\ x=C_1 \cos 2t+C_2 \sin 2t+C_3e^{- \sqrt{6}t}+C_4e^{\sqrt{6}t} + \frac{1}{21} \cos t- \frac{5}{24}t \\
  57. \\ (2): \\
  58. \\ y^{IV}-2y''-24y=2 \cos t+9t \\
  59. \\ A \cos t+ 2A \cos t-24A \cos t=2 \cos t \\
  60. \\ A= - \frac{2}{21} \\
  61. \\ y^{IV}-2y''-24y=9t \\
  62. \\ -24 \left(Ct+D \right)=9t-24Ct-24D=9t->D=0, \ C= - \frac{3}{8} \\
  63. \\ y=C_1 \cos 2t+C_2 \sin 2t+C_3e^{- \sqrt{6}t}+C_4e^{\sqrt{6}t}- \frac{2}{21} \cos t- \frac{3}{8}t
  64. \end{math}
  65. \newpage
  66. Lista 8 zadanie 54 \\
  67. Zbadac stabilnosc rozwiazania zerowego dla nastepujacych ukladow rownan. \\
  68. \begin{math}
  69. \\ X'= \left(
  70. \begin{array}{cc}
  71. 3 & 4 \\
  72. -4 & 3
  73. \end{array}
  74. \right)X \\
  75. \\ \left|
  76. \begin{array}{cc}
  77. 3-l & -4 \\
  78. 4 & 3-l
  79. \end{array}
  80. \right|=0 \\
  81. \\ \left(3-l \right)^2+16=0 \\
  82. \\ 9-6l+l^2+16=0 \\
  83. \\ l^2-6l+25=0 \\
  84. \\ l_1=3-4i, \ l_2=3+4i \\
  85. \\ rel_1=rel_2=3>0 \\
  86. \end{math}
  87. \\ Czesc rzeczywista pierwiastkow rownania charakterystycznego jest dodatnia, wiec punkt rownowagi $(0,0)$ jest niestabilny.
  88. \newpage
  89. Lista 9 zadanie 54 \\
  90. Wyznaczyc rozwiazanie ogolne rownania \\
  91. $a_2(t)y''+a_1(t)y'+a_0(t)y=0$ \\
  92. jezeli $y_1$ jest rozwiazaniem tego rownania. \\
  93. \begin{math}
  94. \\ \left(1+t \right)y''+ty'-y=0, \ y_1(t)=t \\
  95. \\ a_2(t)y''+a_1(t)y'+a_0(t)y=0 \ / \ :a_2(t), \ a_2(t) \neq 0 \\
  96. \\ y''+ \frac{a_1(t)}{a_2(t)}y'+ \frac{a_0(t)}{a_2(t)}y=0 \\
  97. \\ p(t)= \frac{a_1(t)}{a_2(t)}, \ q(t)= \frac{a_0(t)}{a_2(t)} \\
  98. \\ y''+p(t)y'+q(t)=0 \\
  99. y=y_1 \int udt \\
  100. \\ y=y_1 \int udt, \ y'=y_1' \int udt+y_1u, \ y''=y_1'' \int udt+2y_1'+y_1u' \\
  101. \\ y_1u'+[2y_1'+p(x)y_1]u=0 \\
  102. \\ u'+[\frac{2y_1'}{y_1}+p(t)]u=0 \\
  103. \\ u= \frac{C}{y_1^2}e^{- \int p(t)dt}, \ C=1 \\
  104. \\ y_2=y_1 \int \frac{e^{- \int p(t)dt}}{y_1^2}dt \\
  105. \\ y=y_1 \left[C_1+C_2 \int \frac{e^{- \int p(t)dt}}{y_1^2}dt \right] \\
  106. \\ \left(1+t \right)y''+ty'-y=0, \ y_1(t)=t \\
  107. y''+ \frac{t}{1+t}y'- \frac{1}{1+t}y=0, \ t \neq -1 \\
  108. \\ p(t)= \frac{t}{1+t}, \ q(t)= - \frac{1}{1+t} \\
  109. \\ e^{- \int p(t)dt}=e^{- \int \frac{t}{1+t}dt}=e^{-t}(t+1) \\
  110. \\ \int \frac{e^{-t}(t+1)}{t^2}dt= \int e^{-t} \left( \frac{1}{t}+ \frac{1}{t^2} \right) \\
  111. \\ y=t \left[C_1+C_2 \int e^{-t} \left( \frac{1}{t}+ \frac{1}{t^2} \right)dt \right] \\
  112. \end{math}
  113. \newpage
  114. Lista 10 zadanie 31 \\
  115. Wyznaczyc rozwiazanie ogolne rownania niejednorodnego o stalych wspolczynnikach. \\
  116. \begin{math}
  117. \\ y''-2y'+2y=e^t \operatorname{tg} t \\
  118. \\ l^2-2l+2=0 \\
  119. \\ l_1= \frac{2-2i}{2}=1-i, \ l_2=1+i \\
  120. \\u_1=e^t cos t, \ u_2=e^t \sin t \\
  121. \\ u_1'=e^t( \cos t- \sin t), \ u_2'=e^t( \cos t + \sin t) \\
  122. \\ \left|
  123. \begin{array}{cc}
  124. u_1 & u_2 \\
  125. u_1' & u_2'
  126. \end{array}
  127. \right| \left|
  128. \begin{array}{cc}
  129. C_1(t) \\
  130. C_2'(t)
  131. \end{array}
  132. \right| = \left|
  133. \begin{array}{cc}
  134. 0 \\
  135. e^t \operatorname{tg} \end{array}
  136. \right| \\ \\ \left|
  137. \begin{array}{cc}
  138. e^t \cos t & e^t \sin t \\
  139. e^t( \cos t- \sin t) & e^t( \cos t+ \sin t)
  140. \end{array}
  141. \right| \left| \begin{array}{cc}
  142. C_1(t) \\
  143. C_2'(t) \operatorname{tg} \end{array}
  144. \right| \\
  145. \\ W=e^{2t} \cos t( \cos t+\sin t)-e^{2t} \sin t( \cos t- \sin t)= \\
  146. \\ =e^{2t}[\cos^2t+ \cos t \sin t- \sin t \cos t + \sin^2t]= e^{2t}(\cos^2t+ \sin^2t)=e^{2t}>0 \\
  147. \\ W_C_1'= \left|
  148. \begin{array}{cc}
  149. 0 \cos t & e^t \sin t \\
  150. e^t \operatornane{tg} t & e^t \operatorname{tg} t
  151. \end{array}
  152. \right| = -e^{2t} \sin t \operatorname{tg} t= -e^{2t} \frac{ \sin^2t}{ \cos t} \\
  153. W_C_2'= \left|
  154. \begin{array}{cc}
  155. e^t \cos t & 0 \\
  156. e^t(\cos t- \sin t) & e^t \operatorname{tg} t
  157. \end{array}
  158. \right| =e^{2t} \cos t \operatorname{tg} t= e^{2t} \sin t \\
  159. \\ C_1'(t)= \frac{W_C_1'}{W}= \frac{-e^{2t} \sin^2t}{e^{2t} \cos t}= - \frac {\sin^2t}{\cos t} \\
  160. \\ C_2'(t)= \frac{e^{2t} \sint}{e^{2t}}= \sin t \\
  161. \\ C_2(t)= \int \sin tdt= - \cos t+D_2 \\
  162. \\ C_1(t)=- \int \frac{\sin^2t}{\cos t}dt= \int \frac{ \cos^2t-1}{\cos t}dt= \sin t - \ln \left| \operatorname{tg} \frac{x}{2}+ \frac{\pi}{4} \right| + D_1 \\
  163. \\y(t)=C_1(t)U_1(t)+C_2(t)U_2(t)=\left(\sin t- \ln \left| \operatorname{tg} \frac{x}{2}+ \frac{\pi}{4} \right| +D_1 \right)e^t \cos t+ \left(- \cos t +D_2 \right)e^t \sin t \\
  164. \end{math}
  165. \newpage
  166. Lista 11 zadanie 44 \\
  167. Rozwiazac rownanie lub uklad rownan przy uzyciu transformaty Laplace'a \\
  168. \begin{math}
  169. y''+y=te^{-t}+t, \ y(0)=1, \ y'(0)=0 \\
  170. \\ \mathcal{L}[y'']+\mathcal{L}[y]=\mathcal{L}[te^{-t}]+\mathcal{L}[t] \\
  171. \\ z^2\mathcal{L}[y]-z\mathcal{L}[y(0)]-y'(0)+ \mathcal{L}[y]= \frac{1}{(z+1)^2}+ \frac{1}{z^2} \\
  172. \\ \left( z^2+1 \right) \mathcal{L}[y]=zy(0)+y'(0)+ \frac{1}{(z+1)^2}+ \frac{1}{z^2}\\
  173. \\ \mathcal{L}[y]= \frac{z}{z^2+1}+ \frac{1}{(z^2+1)(z+1)^2}+ \frac{1}{(z^2+1)z^2} \\
  174. \\ \frac{1}{(z^2+1)(z+1)^2} = \frac{Az+B}{z^2+1}+ \frac{C}{z+1}+ \frac{D}{z+1} \\
  175. \\ \begin{cases}
  176. A+C=0 \\
  177. 2A+B+C+D=0 \\
  178. A+2B+C=0 \\
  179. B+C+D=1
  180. \end{cases} \ \begin{cases}
  181. A= -\frac{1}{2} \\
  182. B=0 \\
  183. C= \frac{1}{2} \\
  184. D= \frac{1}{2}
  185. \end{cases} \\
  186. \\ \frac{1}{(z^2+1)(z+1)^2} = \frac{-\frac{1}{2}z}{z^2+1}+\frac{\frac{1}{2}}{z+1}+\frac{\frac{1}{2}}{(z+1)^2} \\
  187. \\ \frac{1}{(z^2+1)z^2}= \frac{Az+B}{z^2+1}+ \frac{C}{z}+ \frac{D}{z^2} \\
  188. \\ \begin{cases}
  189. A= -C \\
  190. B=-D \\
  191. C=0 \\
  192. D=1
  193. \end{cases} \ \begin{cases}
  194. A=0 \\
  195. B=-1 \\
  196. C=0 \\
  197. D=1
  198. \end{cases} \\
  199. \\ \frac{1}{(z^2+1)(z+1)^2}= - \frac{1}{z^2+1}+ \frac{1}{z^2} \\
  200. \\ y(t)= \cos t - \frac{1}{2} \cos t + \frac{1}{2}e^{-t}+ \frac{1}{2}te^{-t}- \sin t +t
  201. \end{math}
  202. \end{document}
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