NOUN POP Past Questions: MTH422 - Partial Differential Equation

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JANUARY/FEBRUARY 2013 EXAMINATION

COURSE CODE: MTH 422

COURSE TITLE : PARTIAL DIFFERENTIAL EQUATION

TIME ALLOWED: 3HOURS

FOR WHOM: 400 LEVEL MATHEMATICS, COMPUTER AND MATHEMATICS AND B.ED MATHEMATICS STUDENTS.

INSTRUCTION: ANSWER FOUR FROM SEVEN QUESTIONS. EQUATION ONE IS COMPOUSORY.

1. Given

Find a. The initial element if 5marks

b. The characteristics stripe containing the initial elements 5marks

c. The integral surface which contain the initial element. 4marks

State and Prove CAUCHY KOVALEWASKI Theorem. 14marks

3a. Find the general solution of

,

By method of langrage multiplier 7marks

.3b.. Derive the solution to the Cauchy problem

u_tt=a^2 u_xx+cos〖x,u(x,0)=sin〖x,u_t (x,0)=1+x〗 〗 7marks

Prove that u=F(xy)+xG(y/(x )) is the general solution of x^2 u_xx-y^2 u_yy=014marks

A) Determine the characteristic equation, the characteristic curve and the canonical form of

x^2 u_xx+2xyu_xy+y^2 u_yy+xyu_x+y^2 u_y=0 7marks

5 b) Prove that the equation in 6a above can be solved 7marks

6.. By inspection, classify the following partial differential equations into the foolowing: non-linear, quasi-linear and linear. If linear, determine whether each is homogeneous or not

u_xx+u_yy-2u=x^2

u_x^2+log〖u=〗 2xy

(sin〖u_x)u_x+u_x=e^x 〗

〖2u〗_xx-4u_xy+〖2u〗_yy+3u=0 3.5marks each= 14marks

Solutions2tma@gmail.com

Whatsapp: 08155572788

## NOUN POP Past Questions: MTH422 - Partial Differential Equation

### Re: NOUN POP Past Questions: MTH422 - Partial Differential Equation

Solutions2tma@gmail.com

Whatsapp: 08155572788

MARCH/APRIL 2014 EXAMINATION

COURSE CODE: MTH 422

COURSE TITLE: PARTIAL DIFFERENTIAL EQUATION

TIME ALLOWED: 2Hrs. 30mins

INSTRUCTION: INSTRUCTION: ANSWER ANY FOURQUESTIONS.

INSTRUCTION: ANSWER ANY FOURQUESTIONS. 2Hrs. 30mins

1. Solve the vibration of an elastic string governed by the one dimensional wave equation.

subject to the boundary condition

14marks

2. Given

Find a. The initial element if 5marks

b. The characteristics stripe containing the initial elements 5marks

c.The integral surface which contain the initial element. 4marks

3. State and Prove CAUCHY KOVALEWASKI Theorem. 14marks

4a. Find the general solution of

,

By method of Lagrange multiplier 7marks

4b.. Derive the solution to the Cauchy problem

u_tt=a^2 u_xx+cos〖x,u(x,0)=sin〖x,u_t (x,0)=1+x〗 〗 7marks

5. Prove that u=F(xy)+xG(y/(x )) is the general solution of x^2 u_xx-y^2 u_yy=014marks

6a) Determine the characteristic equation, the characteristic curve and the canonical form of

x^2 u_xx+2xyu_xy+y^2 u_yy+xyu_x+y^2 u_y=0 7marks

6 b) Prove that the equation in 6a above can be solved 7marks

7. By inspection, classify the following partial differential equations into the foolowing: non-linear,quasi-linearand linear. If linear, determine whether each is homogeneous or not

u_xx+u_yy-2u=x^2

u_x^2+log〖u=〗 2xy

(sin〖u_x)u_x+u_x=e^x 〗

〖2u〗_xx-4u_xy+〖2u〗_yy+3u=0

3.5marks each= 14marks

The total obtainable marks is 70 marks. Good Luck.

Solutions2tma@gmail.com

Whatsapp: 08155572788

Whatsapp: 08155572788

MARCH/APRIL 2014 EXAMINATION

COURSE CODE: MTH 422

COURSE TITLE: PARTIAL DIFFERENTIAL EQUATION

TIME ALLOWED: 2Hrs. 30mins

INSTRUCTION: INSTRUCTION: ANSWER ANY FOURQUESTIONS.

INSTRUCTION: ANSWER ANY FOURQUESTIONS. 2Hrs. 30mins

1. Solve the vibration of an elastic string governed by the one dimensional wave equation.

subject to the boundary condition

14marks

2. Given

Find a. The initial element if 5marks

b. The characteristics stripe containing the initial elements 5marks

c.The integral surface which contain the initial element. 4marks

3. State and Prove CAUCHY KOVALEWASKI Theorem. 14marks

4a. Find the general solution of

,

By method of Lagrange multiplier 7marks

4b.. Derive the solution to the Cauchy problem

u_tt=a^2 u_xx+cos〖x,u(x,0)=sin〖x,u_t (x,0)=1+x〗 〗 7marks

5. Prove that u=F(xy)+xG(y/(x )) is the general solution of x^2 u_xx-y^2 u_yy=014marks

6a) Determine the characteristic equation, the characteristic curve and the canonical form of

x^2 u_xx+2xyu_xy+y^2 u_yy+xyu_x+y^2 u_y=0 7marks

6 b) Prove that the equation in 6a above can be solved 7marks

7. By inspection, classify the following partial differential equations into the foolowing: non-linear,quasi-linearand linear. If linear, determine whether each is homogeneous or not

u_xx+u_yy-2u=x^2

u_x^2+log〖u=〗 2xy

(sin〖u_x)u_x+u_x=e^x 〗

〖2u〗_xx-4u_xy+〖2u〗_yy+3u=0

3.5marks each= 14marks

The total obtainable marks is 70 marks. Good Luck.

Solutions2tma@gmail.com

Whatsapp: 08155572788

### Re: NOUN POP Past Questions: MTH422 - Partial Differential Equation

Solutions2tma@gmail.com

Whatsapp: 08155572788

SEPTEMBER/OCTOBER 2015 EXAMINATION

SCHOOL OF SCIENCE AND TECHNOLOGY

COURSE CODE: MTH 422

COURSE TITLE: PARTIAL DIFFERENTIAL EQUATIONTOTALS: 70 MARKS

TIME: 3 HOURS

CREDIT UNIT: 3

INSTRUCTION:ANSWER ANY FIVE QUESTIONS

(a) Using the Methods of Lagrange Multiplier find the general solution of

. 7 marks

(b) Derive the solution to the Cauchy problem

u_tt=a^2 u_xx+cos〖x, u(x,0)=sin〖x,〖 u〗_t (x,0)=1+x〗 〗 7marks

Solve the vibration of an elastic string governed by the one-dimensional wave equation.

Where u(x,t) is the deflection of the string. Since the string is fixed at the ends x =0 and x = L, we have the two boundary conditions thus.

.i.e for all t.

The form of the motion of the string will depend on the initial deflection (deflection at t=0) and on trhe initial velocity (velocity at t=0). Denoting the initial deflection by f(x) and the initial velocity by g(x), the two initial conditions are

or when t=0. 14marks.

(a) Solve , 6marks

(b) Identify the orders and Linearity of these equations.

i)

ii)

iii)

iv)

(2marks each i.e 2*4=8marks)

(a) Consider the function

Calculate and 2marks

Use (a) to show that 2marks

(b) Consider the equation with the following boundary conditions

, and initial condition

Find the solution of the BVP. 10 marks

(a) State and prove Cauchy Kovalewasski theorem. 7marks

(b) Prove that is the general solution of

7marks

Determine the characteristic equations, the characteristic curve and the canonical form of 14marks

Given . Find:

(a).The initial element if 5marks

(b).The characteristics stripe containing the initial elements 5marks

(c).The integral surface which contain the initial element. 4marks

Solutions2tma@gmail.com

Whatsapp: 08155572788

Whatsapp: 08155572788

SEPTEMBER/OCTOBER 2015 EXAMINATION

SCHOOL OF SCIENCE AND TECHNOLOGY

COURSE CODE: MTH 422

COURSE TITLE: PARTIAL DIFFERENTIAL EQUATIONTOTALS: 70 MARKS

TIME: 3 HOURS

CREDIT UNIT: 3

INSTRUCTION:ANSWER ANY FIVE QUESTIONS

(a) Using the Methods of Lagrange Multiplier find the general solution of

. 7 marks

(b) Derive the solution to the Cauchy problem

u_tt=a^2 u_xx+cos〖x, u(x,0)=sin〖x,〖 u〗_t (x,0)=1+x〗 〗 7marks

Solve the vibration of an elastic string governed by the one-dimensional wave equation.

Where u(x,t) is the deflection of the string. Since the string is fixed at the ends x =0 and x = L, we have the two boundary conditions thus.

.i.e for all t.

The form of the motion of the string will depend on the initial deflection (deflection at t=0) and on trhe initial velocity (velocity at t=0). Denoting the initial deflection by f(x) and the initial velocity by g(x), the two initial conditions are

or when t=0. 14marks.

(a) Solve , 6marks

(b) Identify the orders and Linearity of these equations.

i)

ii)

iii)

iv)

(2marks each i.e 2*4=8marks)

(a) Consider the function

Calculate and 2marks

Use (a) to show that 2marks

(b) Consider the equation with the following boundary conditions

, and initial condition

Find the solution of the BVP. 10 marks

(a) State and prove Cauchy Kovalewasski theorem. 7marks

(b) Prove that is the general solution of

7marks

Determine the characteristic equations, the characteristic curve and the canonical form of 14marks

Given . Find:

(a).The initial element if 5marks

(b).The characteristics stripe containing the initial elements 5marks

(c).The integral surface which contain the initial element. 4marks

Solutions2tma@gmail.com

Whatsapp: 08155572788

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