11 Verify that the equation (1 + yexy) dx + (2y + xexy) dy = 0 is exact and then find the general solution.
(9 points)
12 Given the nonhomogeneous system of linear equations: x1 - 3x2 + 10x3 + 4x4 + 4x5 = 2
3x2 - 6x3 - 4x4 - 3x5 = -5
2x1 - 6x2 + 20x3 + 2x4 + 8x5 = -8
Express the solution to the system in vector form if the augmented matrix for the system has an RREF of
[
1 0 −2
0 1 −2
0 0 0
3 0
2 0
0 1
|
−24
−7
4
] (9 points)

Page 5
13 Find the general solution to xy′ + 6y = 3xy4/3 (11 points)
14 Solve the initial value problem y′′ + y = u(t – 𝜋), y(0) = 0, y′(0) = 1 (10 points)

Page 6
15 (A) Consider a spring-mass system with m = 6 kg, c = 3 kg/s, k = 1 N/m and an external force
f(t) = 2 cos ( 𝑡
√3). Find the general solution for the position function x(t) as a function of time t ≥ 0.
(11 points)
(B) Express the steady-periodic part of the solution in amplitude-phase form. (4 points)

Page 7
16 Given matrix A = [
2 0 0
0 2 1
1 0 2
] , find the eigenvalues and corresponding eigenvectors. State the algebraic and
geometric multiplicity of each. (11 points)

Page 8
17 The matrix A = [3 −2
4 −1] has complex eigenvalue λ = 1 + 2i with complex eigenvector [ 1
1 − 𝑖] .
(A) Find the general solution to the homogeneous system X ′ = AX in terms of real-valued functions of t.
(11 points)
(B) Find the solution satisfying x1(0) = 2 and x2(0) = 2

### Problem 11: Verify Exactness and Find the General Solution

Given the differential equation:
\[ (1 + y e^{xy}) \, dx + (2y + x e^{xy}) \, dy = 0 \]

#### Step 1: Verify Exactness
To verify if the given equation is exact, we need to check if:
\[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \]

Here, \( M(x,y) = 1 + y e^{xy} \) and \( N(x,y) = 2y + x e^{xy} \).

Calculate \( \frac{\partial M}{\partial y} \):
\[ \frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (1 + y e^{xy}) = e^{xy} + xy e^{xy} \]

Calculate \( \frac{\partial N}{\partial x} \):
\[ \frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (2y + x e^{xy}) = y e^{xy} + x y e^{xy} = e^{xy}(y + xy) \]

Since \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} = e^{xy}(1 + xy) \), the equation is exact.

#### Step 2: Find the General Solution
The general solution can be found by integrating \( M(x,y) \) with respect to \( x \) and \( N(x,y) \) with respect to \( y \):

\[ \int M(x,y) \, dx = \int (1 + y e^{xy}) \, dx = x + \frac{e^{xy}}{x} + h(y) \]
\[ \int N(x,y) \, dy = \int (2y + x e^{xy}) \, dy = y^2 + \frac{e^{xy}}{y} + g(x) \]

Combining these, we get the potential function \( \psi(x,y) \):
\[ \psi(x,y) = x + y e^{xy} + C \]

Thus, the general solution is:
\[ x + y e^{xy} = C \]

---

### Problem 12: Solve the Nonhomogeneous System in Vector Form

Given the nonhomogeneous system of linear equations with the given augmented matrix RREF:
\[ \begin{bmatrix}
1 & 0 & -2 & 3 & 2 & | & -24 \\
0 & 1 & -2 & 0 & 1 & | & -7 \\
0 & 0 & 0 & 1 & 0 & | & 4 \\
\end{bmatrix} \]

Express the solution in vector form:
Let \( x_3 = t \) and \( x_5 = s \), where \( t \) and \( s \) are free parameters.

From the RREF:
\[ x_1 - 2x_3 + 3x_4 + 2x_5 = -24 \]
\[ x_2 - 2x_3 + x_5 = -7 \]
\[ x_4 = 4 \]

Thus:
\[ x_1 = 2t - 3(4) - 2s - 24 = 2t - 2s - 36 \]
\[ x_2 = 2t - s - 7 \]
\[ x_4 = 4 \]
\[ x_3 = t \]
\[ x_5 = s \]

So, the solution in vector form is:
\[ \mathbf{x} = \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
x_5
\end{bmatrix} = \begin{bmatrix}
2t - 2s - 36 \\
2t - s - 7 \\
t \\
4 \\
s
\end{bmatrix} = t \begin{bmatrix}
2 \\
2 \\
1 \\
0 \\
0
\end{bmatrix} + s \begin{bmatrix}
-2 \\
-1 \\
0 \\
0 \\
1
\end{bmatrix} + \begin{bmatrix}
-36 \\
-7 \\
0 \\
4 \\
0
\end{bmatrix} \]

---

### Problem 13: Find the General Solution

Given:
\[ xy' + 6y = 3x y^{4/3} \]

This is a Bernoulli equation. Rewrite it as:
\[ y' + \frac{6}{x} y = 3 y^{4/3} \]

Let \( v = y^{1/3} \). Then \( y = v^3 \) and \( y' = 3v^2 v' \). Substituting these into the equation:
\[ 3v^2 v' + \frac{6}{x} v^3 = 3 v^4 \]
\[ v' + \frac{2

Explore solutions to exact differential equations, vector forms of nonhomogeneous systems, Bernoulli equation transformations, and spring-mass system general solutions. Includes methods for finding eigenvalues and eigenvectors. Suitable for undergraduate students studying advanced mathematics.

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