This document contains a tutorial on solving various types of differential equations using different methods like method of undetermined coefficients. It includes 10 problems asking students to:
1) Verify particular integrals and find values of constants for given differential equations.
2) Find the general solution using undetermined coefficients method for equations with specific functions on the right side.
3) Solve initial value problems for differential equations with given initial conditions.
4) Find the general solution for various differential equations with combinations of exponential, trigonometric, and polynomial functions on the right side.
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Class 10 Maths contains mainly three concepts one is numbers second is geometry and third is trigonometry and Mensuration. NCERT solutions for class 10 maths are really helpful while doing home work and preparing for exams to score good marks in the board exam. https://www.entrancei.com/topics-topics-class-10-mathematics
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems. We started to develop ways to enhance students IQ. We started to leave an indelible mark on the students who have undergone APEX training. That is why APEX INSTITUTE is very well known of its quality of education
1 of 402.5 PointsUse Cramer’s Rule to solve the following syst.docxmercysuttle
1 of 40
2.5 Points
Use Cramer’s Rule to solve the following system.
x + 2y = 3
3x - 4y = 4
A. {(3, 1/5)}
B. {(5, 1/3)}
C. {(1, 1/2)}
D. {(2, 1/2)}
2 of 40
2.5 Points
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y - z = -2
2x - y + z = 5
-x + 2y + 2z = 1
A. {(0, -1, -2)}
B. {(2, 0, 2)}
C. {(1, -1, 2)}
D. {(4, -1, 3)}
3 of 40
2.5 Points
Use Cramer’s Rule to solve the following system.
2x = 3y + 2
5x = 51 - 4y
A. {(8, 2)}
B. {(3, -4)}
C. {(2, 5)}
D. {(7, 4)}
4 of 40
2.5 Points
Use Cramer’s Rule to solve the following system.
4x - 5y = 17
2x + 3y = 3
A. {(3, -1)}
B. {(2, -1)}
C. {(3, -7)}
D. {(2, 0)}
5 of 40
2.5 Points
Use Cramer’s Rule to solve the following system.
4x - 5y - 6z = -1
x - 2y - 5z = -12
2x - y = 7
A. {(2, -3, 4)}
B. {(5, -7, 4)}
C. {(3, -3, 3)}
D. {(1, -3, 5)}
6 of 40
2.5 Points
Use Cramer’s Rule to solve the following system.
3x - 4y = 4
2x + 2y = 12
A. {(3, 1)}
B. {(4, 2)}
C. {(5, 1)}
D. {(2, 1)}
Reset Selection
7 of 40
2.5 Points
Use Cramer’s Rule to solve the following system.
x + y + z = 0
2x - y + z = -1
-x + 3y - z = -8
A. {(-1, -3, 7)}
B. {(-6, -2, 4)}
C. {(-5, -2, 7)}
D. {(-4, -1, 7)}
8 of 40
2.5 Points
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
3x1 + 5x2 - 8x3 + 5x4 = -8
x1 + 2x2 - 3x3 + x4 = -7
2x1 + 3x2 - 7x3 + 3x4 = -11
4x1 + 8x2 - 10x3+ 7x4 = -10
A. {(1, -5, 3, 4)}
B. {(2, -1, 3, 5)}
C. {(1, 2, 3, 3)}
D. {(2, -2, 3, 4)}
9 of 40
2.5 Points
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y + z = 4
x - y - z = 0
x - y + z = 2
A. {(3, 1, 0)}
B. {(2, 1, 1)}
C. {(4, 2, 1)}
D. {(2, 1, 0)}
10 of 40
2.5 Points
Solve the system using the inverse that is given for the coefficient matrix.
2x + 6y + 6z = 8
2x + 7y + 6z =10
2x + 7y + 7z = 9
The inverse of:
2
2
2
6
7
7
6
6
7
is
7/2
-1
0
0
1
-1
-3
0
1
A. {(1, 2, -1)}
B. {(2, 1, -1)}
C. {(1, 2, 0)}
D. {(1, 3, -1)}
Reset Selection
11 of 40
2.5 Points
Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
2w + x - y = 3
w - 3x + 2y = -4
3w + x - 3y + z = 1
w + 2x - 4y - z = -2
A. {(1, 3, 2, 1)}
B. {(1, 4, 3, -1)}
C. {(1, 5, 1, 1)}
D. {(-1, 2, -2, 1)}
12 of 40
2.5 Points
Use Cramer’s Rule to solve the following system.
x + y = 7
x - y = 3
A. {(7, 2)}
B. {(8, -2)}
C. {(5, 2)}
D. {(9, 3)}
13 of 40
2.5 Points
Use Gaussian elimination to find the complete solution to each system.
x1 + 4x2 + 3x3 - 6x4 = 5
x1 + 3x2 + x3 - 4x4 = 3
2x1 + 8x2 + 7x3 - 5x4 = 11
2x1 + 5x2 - 6x4 = 4
A. {(-47t + 4, 12t, 7t + 1, t)}
B. {(-37t + 2, 16t, -7t + 1, t)}
...
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1. DEPARTMENT OF MATHEMATICAL SCIENCES
FACULTY OF SCIENCE
UNIVERSITI TEKNOLOGI MALAYSIA
SSCM 1703 DIFFERENTIAL EQUATIONS TUTORIAL 4
1. Verify that yp is a particular integral of the following differential equations.
(a) y′′ 3y′ + 2y = x + 4; yp = 1
4 (2x + 11):
(b) y′′ + 9y = 3 cos 2x sin 2x; yp = 1
5 (3 cos 2x sin 2x):
2. Find the values of a; b; c if yp is a particular integral of the differential equations.
(a)
d2y
dx2
dy
dx
+ 12y = 2x; yp = ax + b:
(b)
d2y
dx2 + 6
dy
dx
−4x; yp =
+ 8y = xe
(
ax2 + bx
)
e−4x:
3. By using the method of the undetermined coefficients, find the general solution of the following
equations.
(a)
d2y
dx2 + 5
dy
dx
+ 4y = 3 2x. (b)
d2z
dx2 + 4
dz
dx
= 2x.
(c)
d2z
dt2 + 2
dz
dt
+ 3z = t 2t2. (d) 2
d2y
dx2 + 5
dy
dx
3y = 6x2 2x + 1.
5. Solve the following Initial value problems (IVP’s).
(a)
d2x
dt2 + x = 3 2t2; x = 7 and
dx
dt
= 0 when t = 0.
(b)
d2y
dt2 +
dy
dt
+ 2y = 2t + 4; y = 1 and
dy
dt
= 2 when t = 0.
6. Solve the following Initial value problems (IVP’s).
(a) y′′ 2y′ 8y = 3e4t; y(0) = 0; y′(0) = 3.
(b) z′′ z′ 6z = 3e−2x; z(0) = 0; z′(0) = 2:
7. By using the method of the undetermined method, find the general solution of the following
equations.
(a) y′′ y′ 2y = sin 2x. (b) y′′ + 4y = 1
2 cos 4x.
(c) y′′ 4y′ + 3y = 2 cos x + 4 sin x. (d) y′′ 4y′ + 4y = 4 cos 2x.
8. Find the general solution of the following differential equations.
(a) y′′ + y′ 2y = 2x 40 cos 2x. (b) y′′ 4y = 6 + e2x.
(c) y′′ y′ 2y = 6x + 6e−x. (d) y′′ + 4y = sin 2x cos x.
9. Find the general solution of the following differential equations.
(a) y′′ y = (5x + 1)e3x. (b) y′′ 2y′ 2y = x (ex e−x).
(c) y′′ y′ 2y = ex cos x. (d) y′′ + 5y′ + 6y = e−2x sin 2x.
(e) y′′ + 4y′ + 4y = x2e−2x. (f) y′′ + 2y′ + y = x cos x.
10. Find the general solution of the differential equation
d2y
dx2 + 8
dy
dx
+ 16y = x
(
12 e
−4x)
:
