Intereting Posts

How does $e^{i x}$ produce rotation around the imaginary unit circle?
Existence and Uniqueness of an IVP
Non surjectivity of the exponential map to GL(2,R)
Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$
Where is the world of Imaginary numbers?
Proof involving the infinite number of primes
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
How to Understand the Definition of Cardinal Exponentiation
Question Involving Transitive Elements of a Set
Coproduct of bounded distributive lattices given as lattices of subsets
Prove that $\sqrt{\frac{a}{a+7b}}+\sqrt{\frac{b}{b+7a}}\geq\sqrt{\frac{a}{a+3b}}+\sqrt{\frac{b}{b+3a}}$
zero of vector field with index 0
$n$ and $n^5$ have the same units digit?
If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.
Show that there is no continuous function $f:\mathbb R\to\mathbb R$ such that $f=1_{}$ a.e.

Solve the ODE

$$ \frac{\partial^{2} u }{\partial \eta^{2}} + \frac{\eta}{2\nu} \frac{\partial}{\partial\eta} = 0 $$

The book uses integrating factor = $ e^{\int\frac{\eta}{2\nu} d\eta}$

Can someone explain from here, how to proceed?

- Differential equations that are also functional
- Fourier Series for $|\cos(x)|$
- The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle
- is it correct or not?
- How to solve $y'' + y = -2\sin(x)$?
- Solve the differential equation $x\frac{dy}{dx} + y = x^3y^6$?

I get $ IF = e^\frac{\eta^{2}}{4\nu}$. Multiply this with the ODE. Not sure what to do next? Because the answer the book gives is

$$ \frac{\partial u}{\partial \eta} = A e^{\frac{-\eta^{2}}{4/nu}}$$ and I don’t understand how to get this.

- How to guess a change of variable for an ODE?
- How do we know that we found all solutions of a differential equation?
- Searching for the value of $p_5$
- Grand Prix Race
- Question on reduction of order for linear ODEs
- To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$
- Looking for a logically coherent book for the self-study of differential equations
- Solving $y' = \sqrt{|y|}$
- What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?
- Why it is absolutely mistaken to cancel out differentials?

$$u”(\eta)+\frac{u'(\eta)\eta}{2\nu}=0\Longleftrightarrow$$

Let $u'(\eta)=w(\eta)$, which gives $u”(\eta)=w'(\eta)$:

$$w'(\eta)=-\frac{\eta w(\eta)}{2\nu}\Longleftrightarrow$$

$$\frac{2vw'(\eta)}{w(\eta)}=-\eta\Longleftrightarrow$$

$$\int\frac{2\nu w'(\eta)}{w(\eta)}\space\text{d}\eta=\int-\eta\space\text{d}\eta\Longleftrightarrow$$

$$2\nu\ln(w(\eta))=-\frac{\eta^2}{2}+\text{C}\Longleftrightarrow$$

$$w(\eta)=e^{-\frac{\eta^2-2\text{C}}{4\nu}}\Longleftrightarrow$$

$$w(\eta)=e^{-\frac{\eta^2+\text{C}}{4\nu}}\Longleftrightarrow$$

$$u'(\eta)=e^{-\frac{\eta^2+\text{C}}{4\nu}}\Longleftrightarrow$$

$$u(\eta)=\int e^{-\frac{\eta^2+\text{C}}{4\nu}}\space\text{d}\eta$$

- Dirac Delta or Dirac delta function?
- Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?
- Monty Hall Problem with Five Doors
- How does a calculator calculate the sine, cosine ,tangent using just a number?
- Any equivalent to the Four color theorem for non-planar graphs?
- Generalization of Bertrand's Postulate
- Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$
- Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) could all be solve by the analytical geometry?
- elliptic functions on the 17 wallpaper groups
- Random walk, Cat and mouse
- Existence and Uniqueness of an IVP
- How prove this limit $\displaystyle\lim_{n\to \infty}\frac{n}{\ln{n}}(1-a_{n})=1$
- Under what conditions is a linear automorphism an isometry of some inner product?
- Unbiased estimators for the moments of 2 not independent random variables
- Homology of orientable surface of genus $g$