To determine To solve: The initial value problem y ″ + 3 y ′ + 2 y = { 4 t , if 0 < t < 1 8 , if t > 1 } , y ( 0 ) = 0 , y ′ ( 0 ) = 0 using Laplace transforms. Explanation Formula used: 1. The Laplace transform of the function e a t is 1 s − a . 2. The Laplace transform of the function u ( t − a ) is e − a s s . 3. The Laplace transform of the function 1 s n = t n − 1 ( n − 1 ) ! . Theorem used: Laplace transform of derivatives: The transforms of the first and second derivatives of f ( t ) satisfies the following conditions. L ( f ′ ) = s L ( f ) − f ( 0 ) ⋯ ( 1 ) L ( f ″ ) = s 2 L ( f ) − s f ( 0 ) − f ′ ( 0 ) ⋯ ( 2 ) Where (1) holds if f ( t ) is continuous for all t ≥ 0 and f ′ ( t ) is piecewise continuous on every finite interval on the semi axis t ≥ 0 . Similarly, (2) holds if f and f ′ are continuous for all t ≥ 0 and satisfy the growth restriction and f ″ is piecewise continuous on every finite interval on the semi axis t ≥ 0 . Definition used: Unit step function or Heaviside Function: The unit step function or the Heaviside function denoted by u ( t − a ) is defined as follows. u ( t − a ) = { 0 , if t < a 1 , if t > a } Calculation: The given initial value problem is y ″ + 3 y ′ + 2 y = { 4 t , if 0 < t < 1 8 , if t > 1 } , y ( 0 ) = 0 , y ′ ( 0 ) = 0 . Use the above mentioned definition of the unit step function and rewrite the given term in the right hand side in terms of the Heaviside function as follows. 4 t ( 1 − u ( t − 1 ) ) + 8 u ( t − 1 ) = 4 t + ( 8 − 4 t ) u ( t − 1 ) Therefore, the given initial value problem can be written as shown below. y ″ + 3 y ′ + 2 y = 4 t + ( 8 − 4 t ) u ( t − 1 ) , y ( 0 ) = y ′ ( 0 ) = 0 Take Laplace transform on both sides of y ″ + 3 y ′ + 2 y = 4 t + ( 8 − 4 t ) u ( t − 1 ) , y ( 0 ) = y ′ ( 0 ) = 0 and substitute the values from equation (1) and equation (2) in the given initial value problem as shown below. s 2 Y + 3 s Y + 2 Y = 4 s 2 + ( 8 s − 4 s 2 ) e − s ( s 2 + 3 s + 2 ) Y = 4 s 2 + ( 8 s − 4 s 2 ) e − s ( s + 2 ) ( s + 1 ) Y = 4 s 2 + ( 8 s − 4 s 2 ) e − s Y = 4 s 2 ( s + 2 ) ( s + 1 ) + ( 8 s − 4 s 2 ( s + 2 ) ( s + 1 ) ) e − s Apply partial fractions on the right hand side of the above equation as follows. Consider the term 4 s 2 ( s + 2 ) ( s + 1 ) . 4 s 2 ( s + 2 ) ( s + 1 ) = A s + B s + 2 + C s + 1 + D s 2 4 s 2 ( s + 2 ) ( s + 1 ) = A s ( s + 2 ) ( s + 1 ) + B s 2 ( s + 1 ) + C ( s + 2 ) s 2 + D ( s + 2 ) ( s + 1 ) s 2 ( s + 2 ) ( s + 1 ) 4 = A s ( s + 2 ) ( s + 1 ) + B s 2 ( s + 1 ) + C ( s + 2 ) s 2 + D ( s + 2 ) ( s + 1 ) ⋯ ( 3 ) Substitute s = 0 in the equation (3) and solve as follows. 4 = 2 D 2 = D Substitute s = − 1 in the equation (3) and solve as follows. 4 = C Similarly, substitute s = − 2 in the equation (3) and solve as follows. 4 = − 4 B − 1 = B Equate the coefficient of s 3 in the equation (3) as shown below. 0 = A + B + C ( Put B = − 1 , C = 4 ) 0 = A − 1 + 4 0 = A + 3 A = − 3 Therefore, substitute the values of A , B , C and D in the equation 4 s 2 ( s + 2 ) ( s + 1 ) = A s + B s + 2 + C s + 1 + D s 2 as follows

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Chapter 6.3, Problem 22P

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To solve: The initial value problem y″+3y′+2y={4t, if 0<t<18, if t>1},y(0)=0,y′(0)=0 using Laplace transforms.

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Chapter 6.3, Problem 23P

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Each answer must be justified and all your work should appear. You will be marked on the quality of your explanations. You can discuss the problems with classmates, but you should write your solutions sepa- rately (meaning that you cannot copy the same solution from a joint blackboard, for exam- ple). Your work should be submitted on Moodle, before February 7 at 5 pm. 1. True or false: (a) if E is a subspace of V, then dim(E) + dim(E) = dim(V) (b) Let {i, n} be a basis of the vector space V, where v₁,..., Un are all eigen- vectors for both the matrix A and the matrix B. Then, any eigenvector of A is an eigenvector of B. Justify. 2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1,2,-2), (1, −1, 4), (2, 1, 1)}. 3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal projection onto the orthogonal complement E. (a) The combinations of projections P+Q and PQ correspond to well-known oper- ators. What are they? Justify your answer. (b) Show…

1. True or false: (a) if E is a subspace of V, then dim(E) + dim(E+) = dim(V) (b) Let {i, n} be a basis of the vector space V, where vi,..., are all eigen- vectors for both the matrix A and the matrix B. Then, any eigenvector of A is an eigenvector of B. Justify. 2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1, 2, -2), (1, −1, 4), (2, 1, 1)}. 3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal projection onto the orthogonal complement E. (a) The combinations of projections P+Q and PQ correspond to well-known oper- ators. What are they? Justify your answer. (b) Show that P - Q is its own inverse. 4. Show that the Frobenius product on n x n-matrices, (A, B) = = Tr(B*A), is an inner product, where B* denotes the Hermitian adjoint of B. 5. Show that if A and B are two n x n-matrices for which {1,..., n} is a basis of eigen- vectors (for both A and B), then AB = BA. Remark: It is also true that if AB = BA, then there exists a common…

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The initial value problem y ″ + 3 y ′ + 2 y = { 4 t , if 0 &lt; t &lt; 1 8 , if t &gt; 1 } , y ( 0 ) = 0 , y ′ ( 0 ) = 0 using Laplace transforms.

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To solve: The initial value problem y″+3y′+2y={4t, if 0<t<18, if t>1},y(0)=0,y′(0)=0 using Laplace transforms.

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Chapter 6 Solutions

The initial value problem y ″ + 3 y ′ + 2 y = { 4 t , if 0 < t < 1 8 , if t > 1 } , y ( 0 ) = 0 , y ′ ( 0 ) = 0 using Laplace transforms.