1. Consider the ODE 4⋅y′′+4⋅y′+y=0 with unknown y defined for t>0. It is given that y1(t)=e^(-t/2) is a solution. Find a solution y2 such that y1and y2 are linearly independent. Give your answer in...

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1. Consider the ODE  4⋅y′′+4⋅y′+y=0 with unknown y defined for t>0. It is given that y1(t)=e^(-t/2) is a solution. Find a solution y2 such that y1and y2 are linearly independent. Give your answer in the form of a function rule in terms of t. y2(t)= 2. Find a particular solution of the equation 2y′′−y′−2y=2⋅sin(t) y(t)= 3. Let b be an arbitrary real number and let y1y1 and y2y2 be two functions defined by:  y1(x)=cos(2⋅x) and y2(x)=5−b⋅cos^2(x) For which value of bb is the Wronskian equal to 00? b=  4. Consider the following linear second-order differential equation.  (2⋅t+1)⋅y′′−4⋅(t+1)⋅y′+4⋅y=4⋅t^2+4⋅t+1 It is given that y1(t)=e^2⋅t and y2(t)=−t−1are solutions of the corresponding homogeneous ODE.  Use variation of parameters to find a particular solution of this ODE. ypart(t)= 5. A cable that is suspended at both ends can be described by a solution of the following second-order ODE √y″=a where y is a function of . Here, a is a constant depending on the material of the cable. We impose two initial conditions which express that the deep point of the cable is at [0,b] (so the tangent line at that point is horizontal): y(0)=b, y′(0)=0 Determine the specific solution to this initial value problem. Give your answer in the form y=f(x), where f(x) is an expression in x using the parameters a and b. 6. Consider the following linear second-order differential equation.  (t^2−9)⋅y′′+3t⋅y′+cos(t)⋅y=e^t Please provide the initial conditions under which the existence and uniqueness theorem can be applied. Top of Form y(−3)=4 and y′(−3)=5 y(3)=4 and y′(3)=5 y(8)=4 and y′(8)=5 y(6)=4 and y′(6)=5 y(9)=4and y′(9)=5 Bottom of Form