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- Q5.4 Let Lcg) (^) 4y) =an you? (x) where the be the following: (n-1) nth + + an-ly Aos Air order Suppose you know 9 2x I 4(y) Ly) = sẻ cosxxẻ xe an linear 2x 2x that: Ly()] = 3 xe when when L[Y₂ (x)] = 7e²x sinxe L[y₂ (x)] = 6e ²x cos x when 2x our e (x) (*) + constants Consider differential equation Find a particular solution be a symbolic function of x any decimals. contain any (x) + a₂ y(x) ал 9, (х) = 6 же ча в 4₂(x) = 14e²x cos x уг 2x 22 93 (x) = 54e²x Cos x + 108€ ²³" sinxe to (*). Your answer should and must not1) Find the inner product (f.j) between the following pairs of functions fands. part a) f and are 2x-penodic with F(x) = sinx and g(x)=x for x = (0,2x). part b) f and Ig are 2x-pensdic with F(x) = ((1+i)x and g(x) = x² for x = (0,2%). part c) f and I are 4x-penodic with f(x) = x and g(x) = e* for x = (-2,2). = (1+i)x for x = (0,2x). part d) f and are 2x-penodic with F(x) = x² and g(x) part e) f and I are 10x-periodic with f(x) = Sin (x³ cos(x)) and g(x) = cos (2x sin(x)) for x = (-5,5).5. Let W be the space of polynomials p(x) with real coefficients of degree < 10, satisfying p(0) = p(1) = p'(1) = 0. The dimension of W is (a) 1 (b) 7 (c) 8 (d) 9 (е) 115B Given the inner product space of polynomials with real coefficients, degree5. Consider the function f! NXN → N, defined by f(m,n) = min +1 a) find f(A)= A= {(1,0), (0, 1), (1,1) 3 b) find pre-image of B = {0,13 by f 1.e., F-1 (B) := f `¹ (0) U f `` (1) c) Is f an injective (or one-to-one) function.? Justify answer. d) Is f a surjective (or onto) function. Justify answer.A natural cubic spline is defined by Ja,(x- 1)' + b, (x – 1)² +c, (x – 1) + d, (x – 1)° + b,x² +1 if 01) show that a GP has a limiting sum if 0<1-r < 2 2) by calculating the common ratio show that there is no GP with 1st term 8 and limiting sum 2. 3) a GP has positive 1st term a and has a limiting sum Sinfinite , show that Sinfinite > (1/2 )a. 4) fjind the reange of values of the limiting sum of a GP with i) a = 6 , ii) a = -8, iii) a>0 iv) a<0Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,