Question 1
Solve the initial value problem y" + 52y' + 2y = 0; y(0) = 2, y'(0) = -5.
◦ y = -2 e
t - 3t e
t◦ y = 2 e
-t - 3t e
-t◦ y = 2 e
t - 3t e
t◦ y = 2 e
-t - 5t e
-t◦ y = 2 e
t - 7t e
tQuestion 2
Solve the initial value problem
![](data:image/png;base64, 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)
- 2
![](data:image/png;base64, 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)
= 0, y(2)= 3, y'(1) = -5.
◦ y = 3 -
![](data:image/png;base64, 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)
e
2(1 - e
2t - 4)
◦ y = 3 +
![](data:image/png;base64, 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)
e
2(1 - e
2t)
◦ y = 3 -
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAAmAA0DASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD7g+OPxd1rwD4nFva+KdJ8K6Nb2MN1d3ep+CdT12OLzJJlMk1xaXMMdpEoi5ebC/eO4AGvbrWTzrWGTzY59yBvNiGEfI6ryeD25P1rhPHXgTxV4wvL7T4PF1pY+DtUtGtNQ02TR/NvlVkKP9luhMixbgc/vIZiDnBAIA7mys4dOs4LS2jEVvBGsUcY6KqjAH4AU1tb+v67a/JdXK11b+v69Pm7nzP8Wfg18NvhpGNYtv2a/hvrfhK0MR1S9XTbGG+iRnCs9va/ZGWcIGyQ0sTHBChjjP0T4W8J6H4G0K10Tw3o2n+H9Ftd32fTtLtUtreHcxdtkaAKuWZmOByWJ6mvO/iVpvxM1bxnYnRvDnhTXPCVj5V1Fb6r4mutOmlvFbcskqR6fOrLGQrIu/G4bzkhNvq8RcxIZFVJCBuVG3AHuAcDI98ChK0b3/r+v+HfQlox9FFFIR//2Q==)
e
2(1 - e
4 - 2t)
◦ y = 3 +
![](data:image/png;base64, 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)
e
2(1 - e
4 - 2t)
◦ y = 3 +
![](data:image/png;base64, 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)
e
2(1 - e
2t - 4)