Solutions Manual for Advanced Engineering Mathematics with MATLAB (5th Edition) by Dean G.

Solutions Manual for Advanced Engineering
Mathematics with MATLAB (5th Edition) by
Dean G. Duffy

CHAPTER 1 – FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS**


1. Solve the first-order linear ODE: dy/dx + 2y = 4x
A) y = 2x - 1 + Ce^{-2x}
B) y = 2x - 1 + Ce^{2x}
C) y = 2x - 1
D) y = Ce^{-2x}
Answer: A
Solution: Integrating factor μ = e^{∫2 dx} = e^{2x}. Multiply: e^{2x} dy/dx +
2e^{2x}y = 4x e^{2x} → d/dx (y e^{2x}) = 4x e^{2x}. Integrate: y e^{2x} = ∫4x
e^{2x} dx = 2x e^{2x} - e^{2x} + C. Divide by e^{2x}: y = 2x - 1 + Ce^{-2x}.


2. Which of the following is an exact differential equation?
A) (x + y) dx + (x - y) dy = 0
B) (2x + 3y) dx + (3x + 2y) dy = 0
C) (x^2 + y^2) dx + 2xy dy = 0
D) (y) dx + (x) dy = 0
Answer: B
Solution: For M dx + N dy = 0, exactness requires ∂M/∂y = ∂N/∂x. For B:
M=2x+3y, ∂M/∂y=3; N=3x+2y, ∂N/∂x=3 → exact.


3. Solve the separable ODE: dy/dx = (x^2)/(y^2)
A) y^3 = x^3 + C
B) y^3 = 3x^3 + C
C) y^3 = x^3 + 3C

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, D) y^3 = 3x^3 + 3C
Answer: A
Solution: y^2 dy = x^2 dx → ∫ y^2 dy = ∫ x^2 dx → y^3/3 = x^3/3 + C → y^3 = x^3
+ 3C → y^3 = x^3 + C'.


4. The integrating factor for dy/dx + P(x)y = Q(x) is:
A) e^{∫P(x)dx}
B) e^{∫Q(x)dx}
C) e^{∫P(x)Q(x)dx}
D) ∫P(x)dx
Answer: A


5. Solve the Bernoulli equation: dy/dx + y = xy^3
A) y^{-2} = Ce^{2x} + x + 0.5
B) y^{-2} = Ce^{-2x} + x + 0.5
C) y^{-2} = Ce^{2x} - x + 0.5
D) y^{-2} = Ce^{-2x} - x + 0.5
Answer: D (detailed solution uses substitution v = y^{-2})


6. The general solution to dy/dx = 2y + 3 is:
A) y = Ce^{2x} - 3/2
B) y = Ce^{2x} + 3/2
C) y = Ce^{-2x} - 3/2
D) y = Ce^{x} - 3
Answer: A


7. A tank contains 100 L of brine with 20 kg of salt. Brine with 0.5 kg/L salt
enters at 4 L/min. Well-mixed brine leaves at 4 L/min. The salt amount S(t)
satisfies:
A) dS/dt = 2 - 0.04S

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, B) dS/dt = 4 - 0.04S
C) dS/dt = 2 - 0.4S
D) dS/dt = 0.5 - 0.04S
Answer: A


8. For the ODE dy/dx = x^2 + y^2 with y(0)=1, Euler's method with h=0.1 gives
y(0.1) ≈:
A) 1.0
B) 1.1
C) 1.2
D) 1.5
Answer: B


9. The critical point of dy/dx = x - y is:
A) (0,0)
B) (1,1)
C) (1,0)
D) (0,1)
Answer: A


10. The order of the ODE d^3y/dx^3 + (dy/dx)^4 = sin x is:
A) 1
B) 2
C) 3
D) 4
Answer: C


11. A homogeneous first-order ODE can be written in the form:
A) dy/dx = F(y/x)


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, B) dy/dx = F(xy)
C) dy/dx = F(x/y)
D) dy/dx = F(x+y)
Answer: A


12. The substitution y = vx transforms a homogeneous ODE into:
A) Separable
B) Exact
C) Linear
D) Bernoulli
Answer: A


13. Solve: (y + x) dx + (x) dy = 0
A) xy + x^2/2 = C
B) xy + y^2/2 = C
C) x^2/2 + xy = C
D) y^2/2 + xy = C
Answer: C


14. The ODE dy/dx = (x+2y)/(2x+y) is:
A) Separable
B) Homogeneous
C) Exact
D) Linear
Answer: B


15. The integrating factor for dy/dx - 2y/x = x^2 is:
A) x^{-2}
B) x^{-1}

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