WS DE D For the questions below, L denotes the differential operator as follows LE +...

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WÁS DE D For the questions below, L denotes the differential operator as follows LE + ai dn dtn dn-1 dt-1 1. Prove that the general solution of Ly = 0 can be written as y(t) = ciyi (t) + + CnYn (t), ... +- ...+an, ai ER where y(t), i = 1,...,n are linearly independent functions. (Hint: Show that the solutions to Ly=0 form a vector space. {y}, is actually a basis for that vector space.) 2. Prove that the general solution of Ly = f(t) can be given as y = C₁y₁ (t)+...+ CnYn(t) + Up where y(t), i = 1,...,n are linearly independent functions and y, is a particular (single) solution of Lu= f(t). (Hint: Use linearity of L, i.e., Ly₁ = f, Ly2 = f implies L(31-32) = 0, and then use the previous result.) 3. Solve the Ricatti equation y = (y-t)²+1, y(0) = 2. (Hint: Observe the right-hand side and determine a particular solution of Ricatti equation? Look for polynomial functions of time.) WÁS DE D For the questions below, L denotes the differential operator as follows LE + ai dn dtn dn-1 dt-1 1. Prove that the general solution of Ly = 0 can be written as y(t) = ciyi (t) + + CnYn (t), ... +- ...+an, ai ER where y(t), i = 1,...,n are linearly independent functions. (Hint: Show that the solutions to Ly=0 form a vector space. {y}, is actually a basis for that vector space.) 2. Prove that the general solution of Ly = f(t) can be given as y = C₁y₁ (t)+...+ CnYn(t) + Up where y(t), i = 1,...,n are linearly independent functions and y, is a particular (single) solution of Lu= f(t). (Hint: Use linearity of L, i.e., Ly₁ = f, Ly2 = f implies L(31-32) = 0, and then use the previous result.) 3. Solve the Ricatti equation y = (y-t)²+1, y(0) = 2. (Hint: Observe the right-hand side and determine a particular solution of Ricatti equation? Look for polynomial functions of time.)