In Problems 11–16 verify that the
13.
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First Course in Differential Equations (Instructor's)
- Problem 16 (#2.3.34).Let f(x) = ax +b, and g(x) = cx +d. Find a condition on the constants a, b, c, d such that f◦g=g◦f. Proof. By definition, f◦g(x) = a(cx +d) + b=acx +ad +b, and g◦f(x) = c(ax +b) + d=acx +bc +d. Setting the two equal, we see acx +ad +b=acx +bc +d ad +b=bc +d (a−1)d=(c−1)b That last step was merely added for aesthetic reasons.arrow_forward2. Which of the following is a general solution to the following: x²y" + xy' + (36x² - 1) y (Hint: As discussed in the lecture, use Y, only when J, and J-, are linearly dependent). A. y = c₁J₁(2x) + C₂J_1(2x) 6 B. y = C₁J₁(x) + C₂Y₁(x) 3 3 C. y = c₁₂/₁(6x) + C₂Y₁(6x) 0 D. y = c₁J₁(6x) + c₂] _1(6x) 2arrow_forward10. Determine three linearly independent solutions to the equation y" + 2y" – 3y = 0 of the form y(x) = e"*, where r is a real number. Remember to prove that these solutions are indeed linearly independent.arrow_forward
- 2. Find if y=x +3x-7 and x 21+1. dtarrow_forward8. Iff₁ = e¹*₂f₂=e³x and fi,f2 and f3 are linearly dependent, the f3 could be (A) sin(3x) (B) ex (C) e 12x (D) e3x-e4x (E) In (7x)arrow_forward4. (S.10). Use Gaussian elimination with backward substitution to solve the following linear system: 2.r1 + 12 – 13 = 5, 1 + 12 – 3r3 = -9, -I1 + 12 +2r3 = 9;arrow_forward
- 1. If y = (x + 1/x) (2x-3, then dy/dx will be ? 2. If matrix A is (2 5) (3 4) and f (x) = x2 +4 , what is the answer to f (A)?arrow_forwardty'''+2y''+y'+ty=0 What is the Wronskian of linearly independent solutions?arrow_forward13 Solve the following linear system of DE; x' = Añ. 9x15x2 + 3x3 4x2 + 3x3 O 13arrow_forward
- This is the first part of a two-part problem. Let P=[-: 1 5₁(t) = [(41) 5₂(t) = - sin(4t). a. Show that y₁ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product y(t) = = 0 [1] -4 Enter your answers in terms of the variable t. -4 sin(4t) -4 cos(4t)] ÿ₁ (t) [181-18] b. Show that y₂ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product Enter your answers in terms of the variable t. 04] 32(t) = [-28]|2(t) 181-181arrow_forward2. Given the following 2 x 2 linear system with constant coefficients x' = Ax (H) x= Ax+g(t), (N) where g is not the zero vector. Which of the following statements are true? Justify your answers. A. If , is a solution to (H) and 7, is a solution to (N), then , +27, is a solution to (N). B. If , and 2 are both solutions to (N), then ₁-2 is a solution to (H).arrow_forward5. Solve the following linear system: dX dt with the initial condition - [83] Y(0) = X [2]arrow_forward
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