3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a...
3. Consider the differential equation ty" - (t+1)yy = te2, t> 0. ert is a solution to the corresponding homogeneous (a) Find a value of r for which y = differential equation (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation
2. Consider the differential equation ty" – (t+1)y' +y = 2t2 t>0. (a) Check that yı = et and y2 = t+1 are a fundamental set of solutions to the associated homogeneous equation. (b) Find a particular solution using variation of parameters.
(3) Consider the differential equation ty' + 3ty + y = 0, 1 > 0. (a) Check that y(t) = 1-1 is a solution to this equation. (b) Find another solution (t) such that yı(t) and (t) are linearly independent (that is, wit) and y(t) form a fundamental set of solutions for the differential equation).
(a) Given yı = et is a solution, find another linearly independent solution to the differential equation. ty" – (t + 1)y' + y = 0 (b) Use variation of parameters to find a particular solution to ty" – (t+1)y' +y=ť?,
3. Given that yı(t) = t, y(t) = t, and yz(t) = are solutions to the homogeneous differential equation corresponding to ty" + t'y" – 2ty' + 2y = 2*, t > 0, use variation of parameters to find its general solution.
Consider the differential equation e24 y" – 4y +4y= t> 0. t2 (a) Find T1, T2, roots of the characteristic polynomial of the equation above. 11,12 M (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. yı(t) M y2(t) = M (C) Find the Wronskian of the fundamental solutions you found in part (b). W(t) M (d) Use the fundamental solutions you found in (b) to find functions ui and Usuch...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the...
5. Repeat the same questions in 4.) for the ODE Py"- tt+2)y+(t+2)y2t3, (t>0) (a) Find the general solution of the homogeneous ODE y"- 5y +6y 0. Particularly find yi and (b) Find the equivalent nonhomogeneous system of first order with the chan of variable y (c) Show that (nvand 2( re solutions of the homogeneous system of ODEs (d) Find the variation of parameters equations that have to be satisfic 1 for y(t) vi(t)u(t) + (e) Find the variation of...
need help solving Homework: 4.6 Variation of Parameters Save Score: 0 of 3 pts 3 of 4 (4 complete) HW Score: 70%. 7 of 10 pts X 4.6.23 Question Help Use variation of parameters to find a general solution to the differential equation given that the functions, and y, are linearly independent solutions to the corresponding homogeneous equation fort0 ty' - (+1) +y3+ y el Y=t+1 A general solution is yt)= Enter your answer in the answer box and then...
7. (10 points) Find a particular solution yp(t) to the nonhomogeneous equation ty + y - y = 24t*, t> 0, given the fact that the general solution of the associated homogeneous equation is yn(t) = cit + cat-, C1, C2 E R