A First Course in Differential Equations, 12th
Edition: Complete Solutions Manual latest
2026/2027
Complete Q&A with Rationales for Differential
Equations Success
SECTION A: INTRODUCTION TO DIFFERENTIAL EQUATIONS
(Chapter 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
Answer: B) Second order, linear
,Rationale: A differential equation is classified by order (the
highest derivative present) and linearity. This equation has a
second derivative as the highest order and is linear because the
dependent variable yy and its derivatives appear to the first
power and are not multiplied together . The standard form
an(x)dnydxn+...+a1(x)dydx+a0(x)y=g(x)an(x)dxndny
+...+a1(x)dxdy+a0(x)y=g(x) confirms it is a second-order linear
ODE.
Question 2
Which of the following is a nonlinear differential equation?
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
Rationale: The equation y′=x2+y2y′=x2+y2 is nonlinear
because the dependent variable yy is raised to a power
, (squared) and is not simply to the first power . All other options
are linear because yy and its derivatives appear to the first
power only.
Question 3
A solution of a differential equation is a function that:
A) Does not satisfy the equation
B) Satisfies the equation when substituted into it
C) Is always constant
D) Is always linear
Answer: B) Satisfies the equation when substituted into it
Rationale: A solution of a differential equation is a function
defined on some interval that, when substituted into the equation,
reduces it to an identity. This means the function and its
derivatives make the equation true for all values in the interval.
Question 4
What is the general solution to the differential equation
y′=3x2y′=3x2?
A) y=x3+Cy=x3+C
Edition: Complete Solutions Manual latest
2026/2027
Complete Q&A with Rationales for Differential
Equations Success
SECTION A: INTRODUCTION TO DIFFERENTIAL EQUATIONS
(Chapter 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
Answer: B) Second order, linear
,Rationale: A differential equation is classified by order (the
highest derivative present) and linearity. This equation has a
second derivative as the highest order and is linear because the
dependent variable yy and its derivatives appear to the first
power and are not multiplied together . The standard form
an(x)dnydxn+...+a1(x)dydx+a0(x)y=g(x)an(x)dxndny
+...+a1(x)dxdy+a0(x)y=g(x) confirms it is a second-order linear
ODE.
Question 2
Which of the following is a nonlinear differential equation?
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
Rationale: The equation y′=x2+y2y′=x2+y2 is nonlinear
because the dependent variable yy is raised to a power
, (squared) and is not simply to the first power . All other options
are linear because yy and its derivatives appear to the first
power only.
Question 3
A solution of a differential equation is a function that:
A) Does not satisfy the equation
B) Satisfies the equation when substituted into it
C) Is always constant
D) Is always linear
Answer: B) Satisfies the equation when substituted into it
Rationale: A solution of a differential equation is a function
defined on some interval that, when substituted into the equation,
reduces it to an identity. This means the function and its
derivatives make the equation true for all values in the interval.
Question 4
What is the general solution to the differential equation
y′=3x2y′=3x2?
A) y=x3+Cy=x3+C
