Find a particular solution to y2yy155ett21 ypSolution Solve

Solution

Solve ( d^2 y(t))/( dt^2)+2 ( dy(t))/( dt)+y(t) = (15.5 e^(-t))/(t^2+1): The general solution will be the sum of the complementary solution and particular solution. Find the complementary solution by solving ( d^2 y(t))/( dt^2)+2 ( dy(t))/( dt)+y(t) = 0: Assume a solution will be proportional to e^(lambda t) for some constant lambda. Substitute y(t) = e^(lambda t) into the differential equation: ( d^2 )/( dt^2)(e^(lambda t))+2 ( d)/( dt)(e^(lambda t))+e^(lambda t) = 0 Substitute ( d^2 )/( dt^2)(e^(lambda t)) = lambda^2 e^(lambda t) and ( d)/( dt)(e^(lambda t)) = lambda e^(lambda t): lambda^2 e^(lambda t)+2 lambda e^(lambda t)+e^(lambda t) = 0 Factor out e^(lambda t): (lambda^2+2 lambda+1) e^(lambda t) = 0 Since e^(lambda t) !=0 for any finite lambda, the zeros must come from the polynomial: lambda^2+2 lambda+1 = 0 Factor: (lambda+1)^2 = 0 Solve for lambda: lambda = -1 or lambda = -1 The multiplicity of the root lambda = -1 is 2 which gives y_1(t) = c_1 e^(-t), y_2(t) = c_2 e^(-t) t as solutions, where c_1 and c_2 are arbitrary constants. The general solution is the sum of the above solutions: y(t) = y_1(t)+y_2(t) = c_1 e^(-t)+c_2 e^(-t) t Determine the particular solution to ( d^2 y(t))/( dt^2)+2 ( dy(t))/( dt)+y(t) = (15.5 e^(-t))/(t^2+1) by variation of parameters: List the basis solutions in y_c(t): y_(b_1)(t) = e^(-t) and y_(b_2)(t) = e^(-t) t Compute the Wronskian of y_(b_1)(t) and y_(b_2)(t): (script capital w)(t) = |e^(-t) | e^(-t) t ( d)/( dt)(e^(-t)) | ( d)/( dt)(e^(-t) t)| = |e^(-t) | e^(-t) t -e^(-t) | e^(-t)-e^(-t) t| = e^(-2 t) Let f(t) = (15.5 e^(-t))/(t^2+1): Let v_1(t) = - integral (f(t) y_(b_2)(t))/((script capital w)(t)) dt and v_2(t) = integral (f(t) y_(b_1)(t))/((script capital w)(t)) dt: The particular solution will be given by: y_p(t) = v_1(t) y_(b_1)(t)+v_2(t) y_(b_2)(t) Compute v_1(t): v_1(t) = - integral (15.5 t)/(t^2+1) dt = -7.75 log(t^2+1) Compute v_2(t): v_2(t) = integral 15.5/(t^2+1) dt = 15.5 tan^(-1)(t) The particular solution is thus: y_p(t) = v_1(t) y_(b_1)(t)+v_2(t) y_(b_2)(t) = -7.75 e^(-t) log(t^2+1)+15.5 e^(-t) t tan^(-1)(t) Simplify: y_p(t) = e^(-t) (15.5 t tan^(-1)(t)-7.75 log(t^2+1)) The general solution is given by: y(t) = y_c(t) + y_p(t) = c_1 e^(-t)+c_2 e^(-t) t+e^(-t) (15.5 t tan^(-1)(t)-7.75 log(t^2+1))