true or false If yı(t) and y2(t) are two solutions of the differential equation y2 –...
if y1(t) and y2(t) are two solutions of the differential equation y^2-y'+y=0 then for any constants c1 and c2 c1y1(t)+c2y2(t) is also a solution true or false and why
3. State whether the following statements are true or false? You don't need to explain your answers 1. The functions yı = sin(t) and y2 = sin(21) are linearly independent for all values of t. [2pts) 2. Laplace Transform C is a linear operator. [2pts) e24 3. If A = then et [2pts) = ( 1), ti 4. If we use method of undetermined coefficients to find the particular solution Y (t) for this differential equation y"+y=sint, then Y (t)...
Consider the ODE: Y'" + y' + 2y + 3y = 0. If yı (t) and y2 (t) are two linearly independent solutions to above ODE, then all solutions to it may be written as y(t) = C1 yı(t) + C2 y2(t) for an appropriate choice of the constants C1 and C2 True O False
(y)2 – 2yy" + y2 = 0. Use an exponential ansatz to find two (possibly complex-valued) solutions yi and y2 of the differential equation. Do any of the theorems assure us that cıyı + c2y2 will also be a so- lution of the differential equation? Answer yes or no, either naming the theorem/principle or else briefly explaining why not.
Please help on these HW problems It can be shown that yı = x-2, y2 = x-6 and y3 = 7 are solutions to the differential equation xạy" + 11xy" + 21y' = 0. W(y1, y2, y3) = For an IVP with initial conditions at x = 3, C1yı + C2y2 + c3y3 is the general solution for x on what interval? It can be shown that yı = x-2, y2 = x-7 and y3 = 5 are solutions to...
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.
8. For an equation y',-y'-6y-0 show that yı + y2 and Cyı are also solutions for any constant C where yi - e3t and y2 e2
a) Assume that y1(c) t and y2)te are solutions of the differential equation t2y_ t(t + 2))" + t(t + 2)y-0, t > 0 Do y1(t) and y2() form a fundamental set of solutions of the O.D.E.? C) State the general solution for this O.D.E. a) Assume that y1(c) t and y2)te are solutions of the differential equation t2y_ t(t + 2))" + t(t + 2)y-0, t > 0 Do y1(t) and y2() form a fundamental set of solutions of...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
differential equations 2. (a) Verify that yı = e cos x and y2 = etsin x are solutions of -2y + 2yło on (-00,00). 204 Chapter 5 Linear Second Order Equations (b) Verify that ifc, and are arbitrary constants then y = cre* cos x + cze sinx is a solution of (A) on (-00,00) (c) Solve the initial value problem y" - 2y + 2y = 0, y(0) = 3. y'(O) = -2