Calculus Differential Equations book recommendations

[Hall]

I ordered Differential Equations and Boundary Value Problem ( Computing and Modelling) by Edwards and Penney. There are several things in the book which I don't like

• Too much focus is given to modelling, almost every topic is explained not from mathematical point of view but from application point of view.
• Improved Euler's Method is not well explained, it has been made a kinda gravy of.
• Here is how they do separable differential equations:

$\text{The first-order differential equation}$ $\frac{dy}{dx} = H (x,y)$ is called separable provided that $H(x,y)$
can be written as the product of a function of $x$ and a function of $y$: $\frac{dy}{dx} = g(x) h(y) = g(x)/f(y)$
where $h(y) = 1/f(y)$. In this case the variables $x$ and $y$ can be separated [...] by writing informally the
equation $f(y) dy = g(x) dx$ which we understand to be concise notation for the differential equation $f(y) \frac{dy}{dx} = g(x)$. It is easy to solve this type of differential equation simply by integrating both sides with respect to $x$:
$\int f( y (x) ) \frac{dy}{dx} dx = \int g(x) dx +C$.
I mean to say that, first of all converting $h(y)$ to $1/f(y)$ was really not needed, we could simply take $h(x)$ to the denominator of LHS. And why not to simply integrate $f(y) dy$ and $g(x) dx$, (Prof. Jerison taught to do it that way only) why to take that $dx$ back again? Well, it may be due to some rules in academia or whatever, but I find it not very easy to understand and remember.

• The notation $D_x$ is used quite a lot, like $D_x \left( \int P(x) dx \right) = P(x)$ and has not been explained anywhere that it stands for $\frac{d}{dx}$.

I've decided to change my book. I would like to you to recommend be books on differential equations which would help in self-teaching, books like Late. Prof. Mattuck's Introduction to Analysis, or books of Prof. Strong. Please make sure the books recommended should have following in its content:

1. First-Order Differential Equations (Slope Fields, Method of Separation, Linear-first Order equations, Substitution Method)
2. Numerical Methods and Mathematical models (Euler's Method, RK2, RK4)
3. Linear Equations of Higher Order
4. System of Differential Equations
5. Laplace Transform Methods
6. Eigenvalues method
7. Intro to PDE

And it would be very nice if their shall be answers to problems.

Thank you.

 

[MidgetDwarf]

Most of the intro ODE books (which the topics you listed fall into this category) are similar to the style of Edwards and Penny.

I really liked Ross: Differential Equations, I believe mine is the 2nd edition?. It is a green book and is published by Blaisdell. It is not crowded with diagrams on every page. Explanations are short. I found the Laplace/Inverse Laplace section to be readable, but I liked the presentation in Zill better.

 

[Falgun]

Try Tenenbaum and Pollard. It's a bit verbose but has answers and is less cook-booky.
Or try Coddington which has next to no applications and is very dry (and short). Both coddington and Tenenbaum Pollard don't treat PDEs at all.

MIT OCW has two courses on ODEs one at the same level as you want and one honors course using Birkhoff and Rota which is much more theoretical.

Some lecture notes from Oxford which seem good (I'm currently reading them):
http://www-thphys.physics.ox.ac.uk/people/AlexanderSchekochihin/ODE/

 

[Vanadium 50]

I am not so sure there is a good DE book. I'm not sure there can be. The subject is vert episodic. "You see something like this, try this." Over and over.

 

[MidgetDwarf]

I ordered Differential Equations and Boundary Value Problem ( Computing and Modelling) by Edwards and Penney. There are several things in the book which I don't like

• Too much focus is given to modelling, almost every topic is explained not from mathematical point of view but from application point of view.
• Improved Euler's Method is not well explained, it has been made a kinda gravy of.
• Here is how they do separable differential equations:

$\text{The first-order differential equation}$ $\frac{dy}{dx} = H (x,y)$ is called separable provided that $H(x,y)$
can be written as the product of a function of $x$ and a function of $y$: $\frac{dy}{dx} = g(x) h(y) = g(x)/f(y)$
where $h(y) = 1/f(y)$. In this case the variables $x$ and $y$ can be separated [...] by writing informally the
equation $f(y) dy = g(x) dx$ which we understand to be concise notation for the differential equation $f(y) \frac{dy}{dx} = g(x)$. It is easy to solve this type of differential equation simply by integrating both sides with respect to $x$:
$\int f( y (x) ) \frac{dy}{dx} dx = \int g(x) dx +C$.
I mean to say that, first of all converting $h(y)$ to $1/f(y)$ was really not needed, we could simply take $h(x)$ to the denominator of LHS. And why not to simply integrate $f(y) dy$ and $g(x) dx$, (Prof. Jerison taught to do it that way only) why to take that $dx$ back again? Well, it may be due to some rules in academia or whatever, but I find it not very easy to understand and remember.

• The notation $D_x$ is used quite a lot, like $D_x \left( \int P(x) dx \right) = P(x)$ and has not been explained anywhere that it stands for $\frac{d}{dx}$.

I've decided to change my book. I would like to you to recommend be books on differential equations which would help in self-teaching, books like Late. Prof. Mattuck's Introduction to Analysis, or books of Prof. Strong. Please make sure the books recommended should have following in its content:

1. First-Order Differential Equations (Slope Fields, Method of Separation, Linear-first Order equations, Substitution Method)
2. Numerical Methods and Mathematical models (Euler's Method, RK2, RK4)
3. Linear Equations of Higher Order
4. System of Differential Equations
5. Laplace Transform Methods
6. Eigenvalues method
7. Intro to PDE

And it would be very nice if their shall be answers to problems.

Thank you.

Differential Equations: From Calculus to Dynamical Systems: Second Edition.​

If you search this book on Amazon, you are able to view some of the contents of this book (view function on Amazon). It is more modern than Ross. Look to see if you prefer this to the book you are using.

I am not so sure there is a good DE book. I'm not sure there can be. The subject is vert episodic. "You see something like this, try this." Over and over.

Having only read the first 3 chapters of Arnold's book, I beg to differ. Although from what I know of ODE (very little), it is not complete. Granted this book is for a second course in ODE, and a student needs at the minimum two semesters of Analysis, upper division LA, an intro Topology to start to understand it a bit.
Not recommended for a first exposure to ODE, or someone who lacks at least these courses.

My friend who has taken a differential geometry course, really enjoys this book.

 

[Dr Transport]

Boyce and DiPrima, I had the third edition almost 40 years ago and it si still gong strong way past the 10th I believe...