Math 204 Homework 4.1 1) Determine the largest interval I on which there exists a unique solution to the given initial value problem a. 𝑦

Math 204 Homework 4.1

1) Determine the largest interval I on which there exists a unique solution to the given initial value problem

a. 𝑦^′′+ (tan 𝑥)𝑦 = 𝑒^𝑥, 𝑦(0) = 1, 𝑦^′(0) = 0

b. 𝑥√𝑥 + 1𝑦^′′′− 𝑦^′+ 𝑥𝑦 = 0, 𝑦 (¹₂) = −1, 𝑦^′(¹₂) = −1, 𝑦^′′(¹₂) = 1

2) For the initial value problem

𝑥(𝑥 − 1)𝑦^′′′− 3𝑥𝑦^′′+ 6𝑥²𝑦^′− (cos 𝑥)𝑦 = √𝑥 + 5 𝑦(𝑥_𝑜) = 1, 𝑦^′(𝑥_𝑜) = 0, 𝑦^′′(𝑥_𝑜) = 7

Determine the values of xo and the intervals I containing xo for which theorem 4.1.1 guarantees the existence of a unique solution on I.

3) The two-parameter family 𝑦 = 𝑐₁𝑒^𝑥cos 𝑥 + 𝑐₂𝑒^𝑥sin 𝑥 is a solution of the differential equation 𝑦^′′− 2𝑦^′+ 2𝑦 = 0. Determine whether a member of the family can be found that satisfies the boundary conditions.

a. 𝑦(0) = 1, 𝑦^′(𝜋) = 0 b. 𝑦(0) = 1, 𝑦(𝜋) = −1 c. 𝑦(0) = 0, 𝑦(𝜋) = 0

4) Determine whether the given set of functions is linearly independent on (−∞, ∞) a. {𝑥 , 𝑥² , 4𝑥 − 3𝑥²}

b. {𝑥 , 𝑥² , 𝑥³ , 𝑥⁴} c. {cos 2𝑥 , 1 , cos²𝑥 }

5) Using the Wronskian, verify that the given functions form a fundamental set of solutions of the given differential equation and find a general solution.

a. 𝑥²𝑦^′′+ 𝑥𝑦^′+ 𝑦 = 0; {cos(ln 𝑥), sin(ln 𝑥)}

b. 𝑥³𝑦^′′′+ 6𝑥²𝑦^′′+ 4𝑥𝑦^′− 4𝑦 = 0; {𝑥 , 𝑥⁻² , 𝑥⁻²ln 𝑥}

6) A fundamental set of solutions for the homogeneous differential equation 𝑦^′′′+ 𝑦^′′+ 3𝑦^′− 5𝑦 = 0

is given by {𝑒^𝑥, 𝑒^−𝑥cos 2𝑥, 𝑒^−𝑥sin 2𝑥}. If 𝑦_𝑝 = 𝑥² is a particular solution of the nonhomogeneous differential equation

𝑦^′′′+ 𝑦^′′+ 3𝑦^′− 5𝑦 = 2 + 6𝑥 − 5𝑥² a. Find the general solution of the nonhomogeneous equation.

b. Find the solution that satisfies the initial conditions 𝑦(0) = −1 , 𝑦^′(0) = 1 , 𝑦^′′(0) = −3

b. Use part (a) to find a particular solution of

𝑦^′′− 6𝑦^′+ 5𝑦 = 5𝑥²+ 3𝑥 − 16 − 9𝑒^2𝑥 𝑦^′′− 6𝑦^′+ 5𝑦 = −10𝑥²− 6𝑥 + 32 + 𝑒^2𝑥

Math 204 Homework 4.2

A. In Problems 1 − 3 the indicated function 𝑦₁(𝑥) is a solution of the given differential equation. Use formula (5), as instructed, to find a esecond solution 𝑦₂(𝑥).

1) 6𝑦^′′+ 𝑦^′− 𝑦 = 0 ; 𝑦₁(𝑥) = 𝑒^{𝑥 3⁄}

2) 4𝑥²𝑦^′′+ 𝑦 = 0 ; 𝑦₁(𝑥) = √𝑥 ln 𝑥

3) (1 − 2𝑥 − 𝑥²)𝑦^′′+ 2(1 + 𝑥)𝑦^′− 2𝑦 = 0 ; 𝑦₁(𝑥) = 𝑥 + 1

B. In Problem (4) the indicated function 𝑦₁(𝑥) is a solution of the given associated

homogeneous equation. Use the method of reduction of order to find a seccond solution 𝑦₂(𝑥) of the homogeneous equation and a particular solution of the givin nonhomogeneous equation .

4) 𝑦^′′− 4𝑦^′+ 3𝑦 = 𝑥 ; 𝑦₁(𝑥) = 𝑒^𝑥

A. In Problems 1 − 7 find the general solution of given second –order differential equation.

1) 5𝑦^′′− 10𝑦^′= 0

2) 𝑦^′′− 49𝑦 = 0

3) 𝑦^′′− 8𝑦^′+ 15𝑦 = 0

4) 3𝑦^′′− 5𝑦^′− 2𝑦 = 0

5) 4𝑦^′′− 24𝑦^′+ 9𝑦 = 0

6) 𝑦^′′+ 𝑦^′+ 2𝑦 = 0

7) 𝑦^′′− 4𝑦^′+ 5𝑦 = 0

B. In Problems 1 − 8 find the general solution of given higher –order differential equation.

1) 𝑦^′′′− 𝑦^′′+ 20𝑦^′= 0

2) 𝑦^′′′+ 27𝑦 = 0

3) 𝑦^′′′+ 100𝑦^′= 0

4) 𝑦^′′′+ 5𝑦^′′− 2𝑦^′− 10𝑦 = 0

5) 𝑦^′′′− 6𝑦^′′+ 12𝑦^′− 8𝑦 = 0

6) 𝑦^′′′+ 3𝑦^′′+ 3𝑦^′+ 𝑦 = 0

7) 25𝑦⁽⁴⁾+ 40𝑦^′′+ 16𝑦 = 0

8) 𝑦⁽⁵⁾− 7𝑦⁽⁴⁾− 18𝑦⁽³⁾ = 0

C. In Problems 1 − 3 solve the given IVP.

1) 𝑦^′′− 4𝑦 = 0 𝑦(0) = 1. , 𝑦^′(0) = 5

2) 𝑦^′′+ 25𝑦 = 0 𝑦(0) = 2. , 𝑦^′(0) = 15

3) 𝑦^′′′+ 12𝑦^′′+ 36𝑦^′ = 0. 𝑦(0) = 0 , 𝑦^′(0) = 1 , 𝑦^′′(0) = −7

D. In Problems 1 − 2 solve the given BVP.

1) 𝑦^′′− 2𝑦^′+ 2𝑦 = 0, 𝑦(0) = 1 , 𝑦(𝜋) = 1

2) 𝑦^′′− 10𝑦^′+ 25𝑦 = 0, 𝑦(0) = 1 , 𝑦′(1) = 11

A. In Problems, 1 − 12 find the general solution of given higher –order differential equation.

1) 𝑦^′′− 𝑦^′= −3 + 𝑒^𝑥

2) 9𝑦^′′+ 4𝑦 = 16

3) 𝑦^′′+ 𝑦^′− 6𝑦 = 2𝑥

4) 𝑦^′′− 16𝑦 = 2𝑒^4𝑥

5) 𝑦^′′− 5𝑦^′= 2𝑥³− 4𝑥²− 𝑥 + 6

6) 𝑦^′′′− 6𝑦^′′ = 3 − cos(𝑥)

7) 𝑦⁽⁴⁾− 𝑦^′′ = 4𝑥 + 2𝑥𝑒^𝑥

8) 𝑦^′′+ 𝑦 = 2𝑥 sin(𝑥)

9) 𝑦^′′− 8𝑦^′+ 7𝑦 = 14𝑥²− 3𝑥 + 4 − 24𝑥𝑒^𝑥

10) 𝑦^′′′− 𝑦^′′− 4𝑦^′+ 4𝑦 = 5 − 𝑒^𝑥− 4𝑒^2𝑥

11) 𝑦^′′+ 2𝑦^′− 24𝑦 = 16 − 𝑥𝑒^𝑥− 2𝑒^6𝑥

12) 𝑦^′′+ 2𝑦^′+ 2𝑦 = sin(𝑥) + 3cos(2𝑥)

B. In Problems, 1 − 3 solve the given IVP.

1) 𝑦^′′+ 4𝑦 = −2 𝑦 (^𝜋₈) =¹₂. , 𝑦^′(^𝜋₈) = 2

2) 𝑦^′′′+ 8𝑦 = 2𝑥 − 5 + 8𝑒^−2𝑥 𝑦(0) = −5 , 𝑦^′(0) = 3 , 𝑦^′′(0) = −4

3) 𝑦^′′+ 4𝑦^′+ 4𝑦 = (3 + 𝑥)𝑒^−2𝑥 , 𝑦(0) = 2 , 𝑦′(0) = 5

C. In Problem 1 solve the given IVP in which the input function 𝑔(𝑥) is discontinues.

1) 𝑦^′′− 2𝑦^′+ 10𝑦 = 𝑔(𝑥) , 𝑦(0) = 0 , 𝑦′(0) = 0 , where

𝑔(𝑥) = {20 if 0 ≤ 𝑥 ≤ π 0 if 𝑥 > 𝜋

A. In Problems, 1 − 4 verify that given differential operatar annihilates the indicated functions.

1) 𝐷²+ 64 , 𝑦 = 2 cos 8𝑥 − 5 sin 8𝑥

2) (𝐷 − 2)(𝐷 + 5) , 𝑦 = 𝑒^2𝑥+ 3𝑒^−5𝑥

3) (𝐷 − 1)³ , 𝑦 = 𝑥²𝑒^𝑥

4) 𝐷⁷, 𝑦 = 10 + 2𝑥 + 3𝑥²− 7𝑥⁶+ 1

B. In Problems, 1 − 6 Find a linearly independent functions that are annihilated by the given differential operator.

1) 𝐷²+ 4𝐷

2) 𝐷²− 12𝐷 + 36

3) 𝐷⁴− 12𝐷³ + 35𝐷²

4) 𝐷²+ 6𝐷 + 10

5) 𝐷² − 5𝐷 − 36

6) 3𝐷³− 2𝐷² − 5𝐷

C. In Problem 1- 5 Find a linear differential operator that annihilates the given functions.

1) 𝑥 − 2 + 𝑥²𝑒^−3𝑥

2) (𝑥 − 2 + 𝑥²)𝑒^−3𝑥

3) 9𝑥²− 3𝑥𝑒^4𝑥cos 2𝑥 + sin 7𝑥 + 1

4) (6 − 𝑥𝑒^2𝑥 )²

5) 𝑥⁵(5 + 3𝑥²)

D. In Problems, 1 − 3 Solve the following DE by using undetermined coefficients ( annihilation opproach )

1) 𝑦⁽⁴⁾− 𝑦^′′ = 𝑥 − 1

2) 𝑦^′′+ 4𝑦 = cos 2𝑥

3) 𝑦^′′− 5𝑦^′+ 6𝑦 = 2𝑥𝑒^3𝑥

A. In Problems, 1 − 9 solve each DE by variation of parameters.

1) 𝑦^′′+ 𝑦 = tan 𝑥

2) 𝑦^′′− 𝑦′ = sinh 2𝑥

3) 𝑦^′′− 𝑦 =_𝑒𝑡^2𝑒_+𝑒^𝑡_−𝑡

4) 𝑦^′′+ 𝑦 = cos²𝑥

5) 𝑒^3𝑥𝑦^′′− 9𝑒^3𝑥𝑦 = 9𝑥

6) 𝑦^′′+ 3𝑦^′+ 2𝑦 = sin 𝑒^𝑥

7) 𝑦^′′− 2𝑦^′+ 𝑦 = _1+𝑥^𝑒𝑥₂

8) 4𝑦^′′− 4𝑦^′+ 𝑦 = 𝑒^𝑥⁄2√1 − 𝑥²

9) 𝑦^′′+ 2𝑦^′+ 𝑦 = 𝑒^−𝑥ln 𝑥

B. In Problem, 1 solve the given IVP.

1) 𝑦^′′+ 𝑦 = sec³𝑥 𝑦(0) = 1. , 𝑦^′(0) =¹₂

Math 204 Homework 4.7

A. In Problems, 1 − 9 solve the given DE..

1) 𝑥²𝑦^′′− 2𝑦 = 0

2) 𝑥𝑦^′′+ 𝑦′ = 0

3) 𝑥²𝑦^′′+ 𝑥𝑦^′+ 4𝑦 = 0

4) 𝑥³𝑦^′′′− 6𝑦 = 0

5) 𝑥³𝑦^′′′+ 𝑥𝑦^′− 𝑦 = 0

6) 𝑥²𝑦^′′+ 3𝑥𝑦^′− 4𝑦 = 0

7) 𝑥²𝑦^′′− 4𝑥𝑦^′+ 6𝑦 = ln 𝑥²

8) 𝑥²𝑦^′′+ 𝑥𝑦^′− 𝑦 = ln 𝑥

9) 𝑥²𝑦^′′− 2𝑥𝑦^′+ 2𝑦 = 𝑥⁴𝑒^𝑥

B. In Problems, 1 − 2 solve the given IVP.

1) 𝑥²𝑦^′′− 5𝑥𝑦^′+ 8𝑦 = 8𝑥⁶ 𝑦 (¹₂) = 0. , 𝑦^′(¹₂) = 0 2) 𝑥²𝑦^′′− 3𝑥𝑦^′+ 4𝑦 = 0 𝑦(1) = 5. , 𝑦^′(1) = 3

1. Solve the given system of differential equations by systematic elimination.

𝑑𝑥

𝑑𝑡 = 𝑥 − 4𝑦 𝑑𝑦

𝑑𝑡 = 𝑥 + 𝑦

2. Solve the given system of differential equations by systematic elimination.

(𝐷 + 2)𝑥 + (𝐷 + 1)𝑦 = sin 2𝑡 5𝑥 + (𝐷 + 3)𝑦 = cos 2𝑡

3. Solve the given IVP 𝑑𝑥

𝑑𝑡 = 4𝑥 + 𝑦 𝑑𝑦

𝑑𝑡 = −2𝑥 + 𝑦 𝑥(0) = 1 , 𝑦(0) = 0