A First Course in Differential Equations, 12th
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Edition: Complete Solutions Manual latest = = = = =
2026/2027 =
Complete Q&A with Rationales for Differential= = = = = =
Equations Success = =
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SECTION A: INTRODUCTION TO DIFFERENTIAL EQUATIONS (Ch
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apter 1) = =
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Question 1 = =
Classify the differential equation: d2ydx2+3dydx−5y=0dx2d2y
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+3dxdy−5y=0 =
A) First order, linear
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B) Second order, linear = = =
C) Second order, nonlinear = = =
D) Third order, linear
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Answer: B) Second order, linear
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Rationale: A differential equation is classified by order (the highe
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st derivative present) and linearity. This equation has a second deri
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,vative as the highest order and is linear because the dependent va
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riable yy and its derivatives appear to the first power and are not
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multiplied together. The standard form an(x)dnydxn+...+a1(x)dy
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dx+a0(x)y=g(x)an(x)dxndny +...+a1(x)dxdy+a0(x)y=g(x) confir = =
ms it is a second-order linear ODE.
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Question 2 = =
Which of the following is a nonlinear differential equation?
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A) d2ydx2+2dydx+y=0dx2d2y+2dxdy+y=0 =
B) y′=x2+y2y′=x2+y2 =
C) dydx+3y=exdxdy+3y=ex =
D) d3ydx3−dydx+y=cos xdx3d3y−dxdy+y=cosx =
Answer: B) y′=x2+y2y′=x2+y2
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Rationale: The equation y′=x2+y2y′=x2+y2 is nonlinear because t
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he dependent variable yy is raised to a power
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(squared) and is not simply to the first power. All other options are lin
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ear because yy and its derivatives appear to the first power only.
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Question 3 = =
A solution of a differential equation is a function that:
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, A) Does not satisfy the equation
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B) Satisfies the equation when substituted into it
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C) Is always constant
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D) Is always linear
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Answer: B) Satisfies the equation when substituted into it
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Rationale: A solution of a differential equation is a function defined
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on some interval that, when substituted into the equation, reduces it to
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an identity. This means the function and its derivatives make the equ
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ation true for all values in the interval.
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Question 4 = =
What is the general solution to the differential equation y′
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=3x2y′=3x2? =
A) y=x3+Cy=x3+C =
B) y=6x+Cy=6x+C =
C) y=x3y=x3 =
D) y=3x3+Cy=3x3+C =
Answer: A) y=x3+Cy=x3+C = = =
Rationale: Separable variables: dy=3x2dxdy=3x2dx. Integrating
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both
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Edition: Complete Solutions Manual latest = = = = =
2026/2027 =
Complete Q&A with Rationales for Differential= = = = = =
Equations Success = =
=
SECTION A: INTRODUCTION TO DIFFERENTIAL EQUATIONS (Ch
= = = = = =
apter 1) = =
=
Question 1 = =
Classify the differential equation: d2ydx2+3dydx−5y=0dx2d2y
= = = =
+3dxdy−5y=0 =
A) First order, linear
= = =
B) Second order, linear = = =
C) Second order, nonlinear = = =
D) Third order, linear
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Answer: B) Second order, linear
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Rationale: A differential equation is classified by order (the highe
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st derivative present) and linearity. This equation has a second deri
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,vative as the highest order and is linear because the dependent va
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riable yy and its derivatives appear to the first power and are not
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multiplied together. The standard form an(x)dnydxn+...+a1(x)dy
= = = = =
dx+a0(x)y=g(x)an(x)dxndny +...+a1(x)dxdy+a0(x)y=g(x) confir = =
ms it is a second-order linear ODE.
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=
Question 2 = =
Which of the following is a nonlinear differential equation?
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A) d2ydx2+2dydx+y=0dx2d2y+2dxdy+y=0 =
B) y′=x2+y2y′=x2+y2 =
C) dydx+3y=exdxdy+3y=ex =
D) d3ydx3−dydx+y=cos xdx3d3y−dxdy+y=cosx =
Answer: B) y′=x2+y2y′=x2+y2
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Rationale: The equation y′=x2+y2y′=x2+y2 is nonlinear because t
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he dependent variable yy is raised to a power
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(squared) and is not simply to the first power. All other options are lin
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ear because yy and its derivatives appear to the first power only.
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=
Question 3 = =
A solution of a differential equation is a function that:
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, A) Does not satisfy the equation
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B) Satisfies the equation when substituted into it
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C) Is always constant
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D) Is always linear
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Answer: B) Satisfies the equation when substituted into it
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Rationale: A solution of a differential equation is a function defined
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on some interval that, when substituted into the equation, reduces it to
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an identity. This means the function and its derivatives make the equ
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ation true for all values in the interval.
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=
Question 4 = =
What is the general solution to the differential equation y′
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=3x2y′=3x2? =
A) y=x3+Cy=x3+C =
B) y=6x+Cy=6x+C =
C) y=x3y=x3 =
D) y=3x3+Cy=3x3+C =
Answer: A) y=x3+Cy=x3+C = = =
Rationale: Separable variables: dy=3x2dxdy=3x2dx. Integrating
= = = =
both
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