equação diferencial

\(\[\begin{align} & Solucionando,temos: \\ & y''\text{ }+\text{ }64y\text{ }=\text{ }0 \\ & r{}^\text{2}\text{ }+\text{ }64\text{ }=\text{ }0 \\ & r{}^\text{2}\text{ }=\text{ }-\text{ }64 \\ & y\text{ }=\text{ }c.e{}^\text{a}.cos\text{ }\left( \text{ }b.x\text{ } \right)\text{ }+\text{ }c.e{}^\text{a}.sen\text{ }\left( \text{ }b.x\text{ } \right) \\ & y\text{ }=\text{ }c.e{}^\text{o}.cos\text{ }\left( \text{ }8.x\text{ } \right)\text{ }+\text{ }c.e{}^\text{o}.sen\text{ }\left( \text{ }8.x\text{ } \right) \\ & y\text{ }=\text{ }c.e{}^\text{o}.cos\text{ }\left( \text{ }8x\text{ } \right)\text{ }+\text{ }c.e{}^\text{o}.sen\text{ }\left( \text{ }8x\text{ } \right) \\ & y\text{ }=\text{ }c.1.cos\text{ }\left( \text{ }8x\text{ } \right)\text{ }+\text{ }c.1.sen\text{ }\left( \text{ }8x\text{ } \right)\text{ } \\ & Logo: \\ & y\left( x \right)\text{ }=\text{ }c.cos\text{ }\left( \text{ }8x\text{ } \right)\text{ }+\text{ }c.sen\text{ }\left( \text{ }8x\text{ } \right). \\ & Condicoes:\text{ }y\left( 0 \right)\text{ }=\text{ }2\text{ }e\text{ }y'\left( 0 \right)\text{ }=\text{ }16 \\ & y\left( 0 \right)\text{ }=\text{ }c.cos\text{ }\left( \text{ }8.0\text{ } \right)\text{ }+\text{ }c.sen\text{ }\left( \text{ }8.0\text{ } \right) \\ & 2\text{ }=\text{ }c.cos\text{ }\left( \text{ }0\text{ } \right)\text{ }+\text{ }c.sen\text{ }\left( \text{ }0\text{ } \right) \\ & 2\text{ }=\text{ }c.1\text{ }+\text{ }c.0 \\ & c\text{ }=\text{ }2\text{ } \\ & \\ \end{align}\] \)

\(\[\begin{align} & y'\left( x \right)\text{ }=\text{ }-\text{ }8.c.sen\text{ }\left( \text{ }8x\text{ } \right)\text{ }+\text{ }8.c.cos\text{ }\left( \text{ }8x\text{ } \right) \\ & Logo: \\ & y'\left( 0 \right)\text{ }=\text{ }-\text{ }8.c.sen\text{ }\left( \text{ }8.0\text{ } \right)\text{ }+\text{ }8.c.cos\text{ }\left( \text{ }8.0\text{ } \right) \\ & 16\text{ }=\text{ }-\text{ }8.c.sen\text{ }\left( \text{ }0\text{ } \right)\text{ }+\text{ }8.c.cos\text{ }\left( \text{ }0\text{ } \right) \\ & 16\text{ }=\text{ }-\text{ }8.c.0\text{ }+\text{ }8.c.1 \\ & 16\text{ }=\text{ }8.c \\ & c\text{ }=\text{ }2 \\ & R\to y\text{ }=\text{ }2.cos\text{ }\left( \text{ }8x\text{ } \right)\text{ }+\text{ }2.sen\text{ }\left( \text{ }8x\text{ } \right). \\ & ou \\ & R\to y\text{ }=\text{ }2.\left[ \text{ }cos\text{ }\left( \text{ }8x\text{ } \right)\text{ }+\text{ }sen\text{ }\left( \text{ }8x\text{ } \right)\text{ } \right]. \\ \end{align}\] \)