Intereting Posts

Are these two quotient rings of $\Bbb Z$ isomorphic?
Class Group of Ring of Integers of $\mathbb{Q}$
Every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix.
Measurability of the inverse of a measurable function
Isomorphism of Banach spaces implies isomorphism of duals?
Ultrapower and hyperreals
Calculation of product of all coprimes of number less than itself
Basis of a subset of finitely generated torsion free module
Finding Jordan basis of a matrix ($3\times3$ example)
How to define the $0^0$?
Proof by induction for golden ratio and Fibonacci sequence
Solution to least squares problem using Singular Value decomposition
Find $g$ using euclids algorithm
lim$_{n\rightarrow \infty}\int _{-\pi}^\pi f(t)\cos nt\,dt$
Why is $x^0 = 1$ except when $x = 0$?

What does it mean for a function to be a solution of a differential equation? I think I understand, but I still don’t have a simple or intuitive understanding.

- stability of a linear system
- What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?
- What is the physical meaning of fractional calculus?
- Initial values are lost (diff eq to Transfer function)?
- Finding Weak Solutions to ODEs
- solution of first order differential equation and maximal interval
- $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$
- third-order nonlinear differential equation
- Why does the “separation of variables” method for DEs work?
- What is a general solution to a differential equation?

Solving $x+1=3$ means finding a value for $x$ that satisfies the equation $x+1=3$. In this case, $x=2$ does the job because $2+1=3$.

Now, given a differential equation such as:

$\frac{dy}{dx}= 2x$ ……….(1)

a solution to this equation is a function (call it $y(x)$). Of course not any function will do. The correct function must satisfy the equation we have in (1) above.

for this particular case, y(x) can be equal to $x^2$.

Why? Because the derivative of $x^2$ with respect to $x$ is $\frac{dy}{dx}= 2x$ which is what we have in our equation.

But wait, what about $y(x)=x^2$+5?

In fact, this is another solution. As you can see, in this case we end up with many solutions all of the form:

$y(x)=x^2+k$

where k is a constant. This is because, all of such functions satisfy our differential equation.

Suppose that $\Omega \subset \mathbb{R}^{n+1}$ and that you have some function $F: \Omega \to \mathbb{R}$. Then $F$ defines the differential equation

$$F(x, y, y', …, y^{(n-1)}) = 0$$

A solution to this differential equation is a function $y: (a,b) \to \mathbb{R}$ defined on some open interval (which could be all of $\mathbb{R}$) such that

$$(x, y(x), y'(x), …, y^{(n-1)}(x)) \in \Omega$$

for all $x \in (a,b)$ and

$$F(x, y(x), y'(x), …, y^{(n-1)}(x)) = 0$$

for all $x \in (a,b)$. Since every ordinary differential equation can be written in such a way this defines the notion of solution unambiguously. Intuitively, you can think of this as “$F$ being $0$ on the graph of $y$ and its derivatives.” In practice you usually add some requirements on $F$ and $\Omega$, for example it is usual to require that $\Omega$ is a region, and that $F$ is continuous and invertible around $0$.

To give an example, if you have the equation

$$y''(x) = 2x^3y(x) + y'(x)^2$$

then you can take

$$F(x_1, x_2, x_3, x_4) = 2x_1^3x_2 + x_3^2 – x_4$$

When you write an algebraic equation you are trying to generalise a calculation for say. In a way you are saying what will be the end result if you give certain input or vice-versa.

Similarly when you write a function you are generalising the notion of the end result itself. After defining a function you can opt for any result your wish and find the input required to get that desired result.

In the same way when you define a differential equation you are actually trying to emphasis that in what way the rate of change of a function and its value will merge together to give the result. This kind of approach is useful in study of complex systems where the rate of changes can be measured with time and which helps in understanding the nature of system mathematically.

- A couple of GRE questions
- Nitpicky question about harmonic series
- Must a Hermitian/Kähler Manifold have a complex structure?
- Inner Product Spaces over Finite Fields
- an “alternate derivation” of Poisson summation formula and discrete Fourier transformation
- Find the equation of the plane passing through a point and a vector orthogonal
- Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?
- Every space is “almost” Baire?
- Brouwer transformation plane theorem
- Monos in $\mathsf{Mon}$ are injective homomorphisms.
- Decomposition of $l$ in a subfield of a cyclotomic number field of an odd prime order $l$
- Show that if $E\subset\mathbb{R}$ is a measurable set, so $f:E\rightarrow \mathbb{R}$ is a measurable function.
- Steps in derivative of matrix expression
- A graph of all of mathematics
- When can you switch the order of limits?