[DFL] Prev. Question Analysis

Chapterwise Compilation of BOU MAT1231 Previous Questions¶

📘 Chapter 1: First-Order ODEs¶

152 Term

(a) Define differential equation. Show that the differential equation of circle touch the x-axis at the origin is
\((x^2 - y^2) dy - 2xy dx = 0\). (1+4)
(b) Solve any three of the following equations:
(i) \(\int dy = (y^2 - 1) dx\);
(ii) \(\frac{dy}{dx} = 1 + e^{x - y};\)
(iii) \(\frac{dy}{dx} = \sin(x+y) + \cos(x+y);\)
(iv) \((x^2 + y^2) dy = xy dx.\) (9)

162 Term

(c) Solve: \(x^2(1 + y^2) \frac{dy}{dx} + y^2(x - 1)x = 0\). (4)

172 Term

(c) Solve: \(\sin^{-1} \left(\frac{dy}{dx} \right) = x + y\). (4)
(a) Define homogeneous differential equation with example. Solve the equation
\((6x - 4y + 1) dy = (3x - 2y + 1) dx\). (1+4)
(b) Determine whether the equation \(y \log y \ dx + (x - \log y) \ dy = 0\) is exact. If it is, then solve. (4)
(c) Define Bernoulli’s equation and hence solve: \(\frac{dy}{dx} + \frac{2y}{x} = \frac{y^3}{x^3}\). (5)

182 Term

(a) Define differential equation. Define also order and degree of a differential equation. Solve the differential equation
\((1 + x^2) dy - (1 - y^2) dx = 0\) (2+2+4)
(b) Solve any two:
(i) \(e^{x - y} dx + e^{y - x} dy = 0\)
(ii) \((x^2 - yx^2) dy + (y^2 + xy^2) dx = 0\)
(iii) \((x^2 + y^2) dy = xy \ dx\) (6)
(a) What is integrating factor of differential equation? Solve the following:
(i) \((12y + 4y^3 + 6x^2) dx + 3(x + xy^2) dy = 0\)
(ii) \(x \frac{dy}{dx} + 2y = x^2 \log x\) (1+7)

202 Term

(a) Make the equation homogeneous and find its solution:
\(\frac{dy}{dx} = \frac{2x + 3y + 3}{2x + 3y + 4}\) (6)
(b) Solve the differential equation:
\(\frac{dy}{dx} = \sin(x + y) + \cos(x + y)\) (4)
(c) Solve the equation:
\(x \sqrt{1 + y^2} dx + y \sqrt{1 + x^2} dy = 0\) (4)

📗 Chapter 2: Mathematical Models and Numerical Methods¶

162 Term

(c) A 30 volt electromotive force is applied to an LR series circuit in which the inductance is 0.2 henry and the resistance is 50 ohms. Find the current \(i(t)\) if \(i(0) = 0\). Determine also the current after a long time. (4)

182 Term

(c) A 30 volt electromotive force is applied to an LR series circuit in which the inductance is 0.2 henry and the resistance is 50 ohms. Find the current \(i(t)\) if \(i(0) = 0\). Determine also the current after a long time. (4)
(c) Write a short note on population modeling and equilibrium solution. (4)

202 Term

(c) A 30 volt electromotive force is applied to an LR series circuit in which the inductance is 0.2 henry and the resistance is 50 ohms. Find the current \(i(t)\) if \(i(0) = 0\). Determine also the current after a long time. (4)

📘 Chapter 3: Linear Systems and Matrices¶

Matrices and Gaussian elimination; Row operations; Inverses of matrices and determinants.

From 152 Term¶

4. (a)
If

\[ A = \begin{bmatrix} 2 & -3 & 1 \\ -3 & 1 & 2 \\ 1 & 2 & -3 \end{bmatrix} \]

then find the value of

\[ A^2 - 3A + 9I_3. \]

6

4. (b)
Find the matrix \(X\), such that

\[ \begin{bmatrix} 3 & -4 \\ -3 & -2 \end{bmatrix} X = \begin{bmatrix} -16 & -6 \\ 7 & 2 \end{bmatrix} \]

4

4. (c)
Solve the following system:

\[ \begin{aligned} x + y + z &= 7 \\ x + 2y + 3z &= 16 \\ x + 3y + 4z &= 22 \end{aligned} \]

4

From 162 Term¶

5. (a)
Solve the following equation using inverse of matrix method:

\[ \begin{aligned} 2x - y + 3z &= 9 \\ x + 3y - 2z &= 4 \\ 3x + 2y + 2z &= 10 \end{aligned} \]

5

5. (b)
Find the rank of the matrix

\[ A = \begin{bmatrix} 1 & 2 & 0 & -1 \\ 2 & 6 & -3 & -3 \\ 13 & 10 & -6 & -5 \end{bmatrix} \]

4

5. (c)
Find the value of the determinant

\[ \left| \begin{array}{cccc} 2 & -1 & 2 & 4 \\ -1 & 2 & -1 & 0 \\ 1 & -3 & 2 & 1 \\ -4 & -2 & 3 & 2 \end{array} \right| \]

5

From 172 Term¶

4. (b)
Find the adjoint and inverse of the matrix

\[ A = \begin{bmatrix} 2 & -1 & -1 \\ 1 & -2 & 1 \\ 1 & -1 & 2 \end{bmatrix} \]

4

4. (c)
Solve the following linear equations with help of matrices:

\[ \begin{aligned} x + 2y + 3z + 4 &= 0 \\ 2x + 4y + 5z + 7 &= 0 \\ 3x + 5y + 6z + 10 &= 0 \end{aligned} \]

5

From 182 Term¶

3. (c)
Prove that

\[ \begin{vmatrix} x & 1 & 1 & \cdots & 1 \\ 1 & x & 1 & \cdots & 1 \\ 1 & 1 & x & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & x \end{vmatrix} = (x - 1)^{n - 1}(x + n - 1) \]

where given determinant is of order \(n\).
4

4. (b)
Define rank of a matrix. Find the rank of the matrix

\[ A = \begin{bmatrix} 1 & 3 & -2 & 5 & 4 \\ 1 & 4 & 1 & 3 & 5 \\ 1 & 4 & 2 & 4 & 3 \\ 2 & 7 & -3 & 6 & 13 \end{bmatrix} \]

4

From 202 Term¶

5. (a)
Define rank of a matrix. Find the rank of matrix

\[ A = \begin{bmatrix} 3 & -2 & 0 & -1 \\ 0 & 2 & 2 & 1 \\ 1 & -2 & 3 & 2 \\ 0 & 1 & 2 & 1 \end{bmatrix} \]

4

5. (b)
Using Cramer’s rule solve:

\[ \begin{aligned} x - y + z &= 1 \\ x + y - 2z &= 0 \\ 2x - y - z &= 0 \end{aligned} \]

5

4. (c)
Write down two matrices \(A\) and \(B\) of order 3×3 and then prove that

\[ (AB)^T = B^T A^T \]

4

📘 Chapter 4: Vector Spaces¶

Vector spaces and subspaces; Linear independence; Bases and dimension; Row and column spaces for matrices.

From 152 Term¶

6. (a)
Show whether the following vectors are a basis of \(\mathbb{R}^3\) or not:

\[ (1, 2, 1),\; (2, 1, 0),\; (1, -1, 2) \]

7

6. (b)
Find a basis and the dimension of the solution space \(W\) of the following homogeneous system:

\[ \begin{aligned} x + 2y + z - 2t &= 0 \\ 2x + 4y + 4z - 3t &= 0 \\ 3x + 6y + 7z - 4t &= 0 \end{aligned} \]

4

6. (c)
Define row space, column space and null space of a matrix.
3

From 162 Term¶

6. (a)
Define a vector space and subspace with example.
4

6. (b)
Show that

\[ S = \{ (a, b, c, d) \in \mathbb{R}^4 : 2a - 3b + 5c - d = 0 \} \]

is a subspace of \(\mathbb{R}^4\)
5

6. (c)
Prove that the vector space \(V\) is the direct sum of its subspaces \(U\) and \(W\) if and only if:
(i) \(V = U + W\); (ii) \(U \cap W = \{0\}\)
5

From 172 Term¶

5. (b)
Define vector space and subspace with example.
4

5. (c)
Determine whether the following sets are subspace of \(\mathbb{R}^3\) or not:
(i) \(S = \{(x, y, 0): x, y \in \mathbb{R} \}\)
(ii) \(T = \{(x, y, 1): x, y \in \mathbb{R} \}\)
6

From 182 Term¶

7. (a)
Define linear dependence and linear independence. Test whether the following vectors are linearly independent or dependent:

\[ (1, -2, 4, 1),\; (2, 1, 0, -3),\; (3, -6, 1, 4) \]

5

7. (b)
Define subspace. Show that

\[ S = \{ (a, b, c, d) \in \mathbb{R}^4 : 2a - 3b + 5c - d = 0 \} \]

is a subspace of \(\mathbb{R}^4\)
1+4

Great! Let's continue with:

📘 Chapter 5: Higher-Order Linear ODEs¶

Second order linear equations; General solution of linear equations; Equations with constant coefficients; Mechanical vibrations.

From 152 Term¶

5. (a)
Transform the following differential equation into an equivalent system of first-order differential equations:

\[ x'' + 3x' + 7x = t^2 \]

7

From 162 Term¶

3. (a)
Define Bernoulli and Riccati equations. Solve

\[ \frac{dy}{dx} + \sin 2y = x^2 \cos^2 y \]

2+4

3. (b)
Find the particular solution of

\[ (D^2 + 4y) = 12x,\quad \text{when } y(0) = 5,\ y'(0) = 7. \]

4

3. (c)
A 30V electromotive force is applied to an LR series circuit in which the inductance is 0.2 henry and the resistance is 50 ohms. Find the current \(i(t)\) if \(i(0) = 0\). Determine also the current after a long time.
4

From 172 Term¶

3. (b)
Find the particular solution of

\[ \frac{d^2y}{dx^2} - \frac{dy}{dx} - 6y = 8e^{2x} - 5e^{3x},\quad \text{when } y(0) = 3,\ y'(0) = 5. \]

6

From 182 Term¶

6. (a)
Write down the common source of mechanical vibration. Solve the following differential equation:

\[ (3x + 2) \frac{d^2y}{dx^2} + 5(3x + 2) \frac{dy}{dx} - 3y = x^2 + x + 1. \]

7

📘 Chapter 6: Eigenvalues and Eigenvectors¶

Eigenvalues and eigenvectors; Diagonalization of matrices; Applications.

From 152 Term¶

7. (a)
Find the eigenvalues of the matrix

\[ A = \begin{bmatrix} 2 & 1 & 0 \\ 3 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]

5

7. (b)
Find a matrix \(P\) that diagonalizes the matrix

\[ A = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix} \]

9

From 162 Term¶

7. (a)
Define eigenvalues and eigenvectors of a square matrix.
3

7. (b)
Find the eigenvalues and the associated eigenvectors of the following matrix:

\[ A = \begin{bmatrix} 8 & 2 & -2 \\ 3 & 3 & -1 \\ 24 & 8 & -6 \end{bmatrix} \]

7

From 172 Term¶

7. (a)
Find the eigenvalues and eigenvectors of the matrix

\[ A = \begin{bmatrix} 4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -4 & -3 \end{bmatrix} \]

Also find the matrix \(P\) that diagonalizes \(A\) and determine \(P^{-1}AP\).
7+3

From 182 Term¶

7. (a)
Define eigenvalues and eigenvectors of a square matrix. Find all the eigenvalues and associated eigenvectors of the matrix:

\[ A = \begin{bmatrix} 1 & 0 & -2 \\ 0 & 0 & 0 \\ -2 & 0 & 4 \end{bmatrix} \]

2+5

From 202 Term¶

6. (b)
Determine Eigenvalues and Eigenvectors for the matrix

\[ B = \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \]

6

📘 Chapter 7: Homogeneous Linear Systems of ODEs¶

Linear systems of differential equations; Homogeneous systems; General solution using eigenvalues/eigenvectors.

From 152 Term¶

5. (b)
Find the general solution of the system

\[ x' = y;\quad y' = 2x + y. \]

7

📘 Chapter 8: Inhomogeneous Linear Systems of ODEs¶

Nonhomogeneous linear systems; Particular solution; Application to modeling.

⚠️ No directly labeled questions found so far that involve full inhomogeneous systems of ODEs in matrix form.

Will include this section if matched in later sets.

📘 Chapter 9: Nonlinear Systems¶

Phase plane analysis; Nonlinear systems; Population models.

From 202 Term¶

6. (c)
Write a short note on Population Modeling and Equilibrium Solution.
4