Calculus HW

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Miran Anwar, mth 235 ss21 1: Hw20-6.1-BVP-SEP. Due: 04/26/2021 at 11:00pm EDT.

See in LN, § 6.1, See Examples 6.1.2-6.1.3. 1. (10 points)

Consider the BVP for the function y given by

y′′+ 25 y = 0, y(0) = 4, y (

) = 5.

(a) Find r1, r2, roots of the characteristic polynomial of the equation above.

r1,r2 =

(b) Find a set of real-valued fundamental solutions to the differential equation above.

y1(x) =

y2(x) =

y(x) =

Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

See in LN, § 6.1, See Examples 6.1.2-6.1.3. 2. (10 points)

Consider the BVP for the function y given by

y′′+ 36 y = 0, y(0) = 1, y(π) = 1.

(a) Find r1, r2, roots of the characteristic polynomial of the equation above.

r1,r2 =

(b) Find a set of real-valued fundamental solutions to the differential equation above.

y1(x) =

y2(x) =

y(x) = 1

Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

See in LN, § 6.1, See Examples 6.1.2-6.1.3. 3. (10 points)

Consider the BVP for the function y given by

y′′+ 9 y = 0, y(0) =−2, y (

) =−3.

(a) Find r1, r2, roots of the characteristic polynomial of the equation above.

r1,r2 =

(b) Find a set of real-valued fundamental solutions to the differential equation above.

y1(x) =

y2(x) =

y(x) =

Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

See in LN, § 6.1, See Examples 6.1.2-6.1.3. 4. (10 points)

Consider the BVP for the function y given by

y′′+ 9 y = 0, y (

) =−5, y

(7π 6

) = 5.

(a) Find r1, r2, roots of the characteristic polynomial of the equation above.

r1,r2 =

(b) Find a set of real-valued fundamental solutions to the differential equation above.

y1(x) =

y2(x) =

y(x) =

Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

See in LN, § 6.1, See Examples 6.1.2-6.1.3. 5. (10 points)

Consider the BVP for the function y given by

y′′+ 9 y = 0, y(0) =−5, y′ (4π

) =−2.

(a) Find r1, r2, roots of the characteristic polynomial of the equation above.

r1,r2 =

(b) Find a set of real-valued fundamental solutions to the differential equation above.

y1(x) =

y2(x) =

y(x) =

Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

See in LN, § 6.1, See Examples 6.1.2-6.1.3. 6. (10 points)

Consider the BVP for the function y given by

y′′+ 25 y = 0, y′(0) =−5, y′ (2π

) =−5.

(a) Find r1, r2, roots of the characteristic polynomial of the equation above.

r1,r2 =

(b) Find a set of real-valued fundamental solutions to the differential equation above.

y1(x) = 3

y2(x) =

y(x) =

Note 1: If there are no solutions, type No Solution. Note 2: If there are infinitely many solutions, use k for the arbitrary constant.

See in LN, § 6.1, See Examples 6.1.4-6.1.5. 7. (10 points)

Find the eigenvalues and eigenfunctions for the differential operator L(y) = −y′′ with boundary conditions y(0) = 0 and y(3) = 0, which is equivalent to the following BVP

y′′+ λ y = 0, y(0) = 0, y(3) = 0.

(a) Find all eigenvalues λn as function of a positive integer n > 1.

λn =

(b) Find the eigenfunctions yn corresponding to the eigenvalues λn found in part (a).

yn(x) =

See in LN, § 6.1, See Examples 6.1.4-6.1.5. 8. (10 points)

Find the eigenvalues and eigenfunctions for the differential operator L(y) = −y′′ with boundary conditions y(0) = 0 and y′(4) = 0, which is equivalent to the following BVP

y′′+ λ y = 0, y(0) = 0, y′(4) = 0.

(a) Find all eigenvalues λn as function of a positive integer n > 1.

λn =

(b) Find the eigenfunctions yn corresponding to the eigenvalues λn found in part (a).

yn(x) =

See in LN, § 6.1, See Examples 6.1.4-6.1.5. 9. (10 points)

Find the eigenvalues and eigenfunctions for the differential operator L(y) = −y′′ with boundary conditions y′(0) = 0 and y(2) = 0, which is equivalent to the following BVP

y′′+ λ y = 0, y′(0) = 0, y(2) = 0.

(a) Find all eigenvalues λn as function of a positive integer n > 1.

λn =

(b) Find the eigenfunctions yn corresponding to the eigenvalues λn found in part (a).

yn(x) =

See in LN, § 6.1, See Examples 6.1.4-6.1.5. 10. (10 points)

Find the eigenvalues and eigenfunctions for the differential operator L(y) = −y′′ with boundary conditions y′(0) = 0 and y′(5) = 0, which is equivalent to the following BVP

y′′+ λ y = 0, y′(0) = 0, y′(5) = 0.

(a) Find all eigenvalues λn as function of a positive integer n > 1.

λn =

(b) Find the eigenfunctions yn corresponding to the eigenvalues λn found in part (a).

yn(x) =

See in LN, § 6.1, See Examples 6.1.4-6.1.5. 11. (10 points)

Note: You have only 5 attempts to solve this problem.

In the picture below we graph the function yα(x) = sin (

απx 4

) . Move the slider for the constant α in the

graph and find the first 5 values of α so that the function yα is a solution of the eigenfunction problem

y′′(x) + λ y(x) = 0, y(0) = 0, y(4) = 0.

α =

Note: Your answer should be a list of 5 numbers separated by commas.

Comments on the graph below: • The first time running it may take a few minutes to load.

• The graph is interactive. • You can move the slider for the constant alpha, (α).

1 See in LN, § 6.1, See Examples 6.1.4-6.1.5. 12. (10 points)

Note: You have only 5 attempts to solve this problem.

In the picture below we graph the function yα(x) = sin ((

απ

)( x 4

)) . Move the slider for the constant α in the

graph and find the first 5 values of α so that the function yα is a solution of the eigenfunction problem

y′′(x) + λ y(x) = 0, y(0) = 0, y′(4) = 0.

α =

Note: Your answer should be a list of 5 numbers separated by commas.

Comments on the graph below: • The first time running it may take a few minutes to load. • The graph is interactive. • You can move the slider for the constant alpha, (α). • The red dasshed line is tangent to the graph of yα(x) at x = 4.