In Problems 35–46 find the general solution of the given system.
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Differential Equations with Boundary-Value Problems (MindTap Course List)
- 13 Solve the following linear system of DE; x' = Añ. 9x15x2 + 3x3 4x2 + 3x3 O 13arrow_forwardProblem 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_forward5. Find the general solution of the given system. X' = [; x. Х.arrow_forward
- Section 2.2 2.1. Solve the following difference equations: (a) Yk+1+Yk = 2+ k, (b) Yk+1 – 2Yk k3, (c) Yk+1 – 3 (d) Yk+1 – Yk = 1/k(k+ 1), (e) Yk+1+ Yk = 1/k(k+ 1), (f) (k + 2)yk+1 – (k+1)yk = 5+ 2* – k2, (g) Yk+1+ Yk = k +2 · 3k, (h) Yk+1 Yk 0, Yk = ke*, (i) Yk+1 Bak? Yk (j) Yk+1 ayk = cos(bk), (k) Yk+1 + Yk = (-1)k, (1) - * = k. Yk+1 k+1arrow_forwardIf L(x)=mx+b is the linearization of the cube root of 3x+1 at x=333 , then b=arrow_forwardProblem 2. What solution of the linear system of differential equations x = x1 – 3.x2 x2 = 3x1 + x2 satisfies the initial conditions X1(0) = 4, X2(0) = 7?arrow_forward
- 5. The following sets of simultaneous equations may or may not be solvable by the Gaussian Elimination method. For each case, explain why. If solvable, solve. (a) (b) (c) (d) x+y+3z=5 2x + 2y + 2z = 14 3x + 3y+9z = 15 2 -1 1] 4 1 3 2 12 3 2 3 16 2x-y+z=0 x + 3y + 2z=0 3x + 2y + 3z == 0 x₁ + x₂ + x3-X₂ = 2 x1-x₂-x₂ + x₁ = 0 2x₁ + x₂-x3 + 2x4 = 9 3x₁ + x₂ + 2x3-X4 = 7arrow_forwardProblem 2. Let r(x) be a rational function given by: ao + a₁x + bo + b₁x + (1) ..." Assume we would like to interpolate a function f with r, satisfying interpolation conditions of the form f(x₁) = r(x₁). Write down a linear system which can be solved to find the parameters ao, a₁, . . . , am, bo, b₁, . . . , bn for a given rational interpolation problem. r(x) = = • amxm bn xnarrow_forwardIn Problems 39–46, show that (f ° g) (x) = (g° f) (x) = x. %3D 39. f(x) = 2x; g(x) = 40. f(x) = 4x; g(x) = i* 41. f(x) = x; g(x) %3! %3D 43. f(x) = 2x – 6; 8(x) = ; (x + 6) 46. fl+) = s(*) = 42. f(x) = x + 5; g(x) = x - 5 44. f(x) = 4 – 3x; g(x) = (4 - x) %3D 45. f(x) = ax + b; g(x) = - (x - b) a + 0 %3D aarrow_forward
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