Appendix A Hints and Answers to Selected Exercises
1 A First Look at Differential Equations
1.1 Modeling with Differential Equations
1.1.9 Exercises
1.1.9.31.
1.4 Analyzing Equations Numerically
1.4.6 Exercises
1.4.6.5.
Hint.
This equation is a first-order linear equation (Section 1.5), but it is possible to find the analytic solution using Sage (Subsection 1.2.10).
1.4.6.8.
1.5 First-Order Linear Equations
1.5.8 Exercises
1.5.8.21.
Hint.
If is the amount of salt in the tank at time we know that The volume of the tank is We can model the amount of salt in the tank at time with a differential equation,
The resulting equation
is a first order linear differential equation. An integrating factor for this equation is given by
Multiplying both sides of the differential equation by we have
Integrating both sides of this equation, we obtain
Using the intial condition we can determine that or
The tank is full at time and the tank contains kilograms of salt when the tank is full.
1.5.8.25. Exact Differential Equations.
1.5.8.28.
1.6 Existence and Uniqueness of Solutions
1.6.5 Exercises
1.6.5.1.
Hint.
-
There exists a unique solution to
since and are continuous at the point -
The Existence and Uniqueness Theorem does not apply to
since is not continuous at -
There exists a unique solution to
since and are both continuous at the point -
The Existence and Uniqueness Theorem does not apply to
since is not continuous at -
There exists a unique solution to
since and are both continuous at the point -
There exists a unique solution to
since and are both continuous at the point -
The Existence and Uniqueness Theorem does not apply to
since is not continuous at
1.6.5.3.
3 Linear Systems
3.2 Planar Systems
3.2.6 Exercises
3.2.6.10.
4 Second-Order Linear Equations
4.1 Homogeneous Linear Equations
4.1.6 Exercises
4.1.6.32.
4.1.6.33.
4.1.6.38. Euler Equations.
4.1.6.39. Higher Order Linear Equqtions with Constant Coefficients.
4.2 Forcing
4.2.7 Exercises
4.2.7.25.
Hint.
Suppose that that and are linearly dependent on an interval Then one function is a multiple of the other, say Thus,
Conversely, suppose that
for all in If then and the two functions are linearly dependent. Assume that for some in Since is differentiable, it must also be continuous and there is some interval contained in such that and does not vanish on this interval. Therefore,
and is constant on the interval Thus, and Since and are both solutions to the differential equation and have the same initial condition, for all by the existence and uniqueness theorem. Consequently, and are linearly dependent.
4.2.7.26.
4.2.7.27.
Hint.
-
If we solve the system
and we obtain -
Integrate the two equations from part (2).
-
The general solution to the homogeneous equation
is