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# Find a linear differential equation satisfied by given equations

In each of the following cases find a second-order linear differential equation satisfied by the given and .

1. ,
2. ,
3. .
4. .
5. .

1. We compute the first two derivatives of the given equations, So, if and are solutions of a linear second order differential equation, then we have and Together these imply Therefore, are solutions of 2. We compute the first two derivatives of the given equations, So, if and are solutions of a linear second order differential equation, then we have and Therefore, are solutions of 3. Rather than take derivatives of these two functions (which gets messy) we consider the function Then if we let and we have . Since this gives us . By Theorem 8.7, then, is a solution of for every choice of and . The functions and correspond to the choices and , respectively. Hence, and are solutions of the second order differential equation 4. Similar to our strategy in part (c), let Letting and we obtain . So these are solutions of the equation . Then, since we have and are solutions to the equation corresponding the choices and , respectively. Therefore and are solutions of 5. First, using the definitions of the hyperbolic sine and cosine we have So if we let with and we have . Therefore, is the set of solutions to . Since and are the the solutions with and , respectively we have and solutions 