Intereting Posts

Communication complexity example problem
If $r$ is a primitive root of odd prime $p$, prove that $\text{ind}_r (-1) = \frac{p-1}{2}$
contractible and simply connected
Limit value of a product martingale
Poker game full house
To what extent can values of $n$ such that $n^2-n+41$ is composite be predicted?
$K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded.
Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places
Parallel vector fields imply a flat connection in constant curvature
Integral of a positive function is positive?
Non-Symmetric Positive Definite Matrices
factorization of a^n+1?
Constructor theory distinguishability
If coercive in a subspace implies coercive in a bigger space under these conditions?
How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$

Does the equation

$$\frac{d^2y}{dx^2} – \frac{H(x)}{b} y = c H(x)$$

have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant?

- Tensor products of functions generate dense subspace?
- When is the difference of two convex functions convex?
- The preimage of continuous function on a closed set is closed.
- smooth functions or continuous
- DE solution's uniqueness and convexity
- Show that $\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$

**Update: What about the second step function be $H(-x)$:**

$$\frac{d^2y}{dx^2} – \frac{H(x)}{b} y = c H(-x)$$

- Counterexamples in Double Integral
- Defining a metric space
- How can I know the time difference between two cities almost at the same latitude?
- $f'(x) = g(f(x)) $ where $g: \mathbb{R} \rightarrow \mathbb{R}$ is smooth. Show $f$ is smooth.
- Electricity differential equation problem
- When can a sum and integral be interchanged?
- Calculate $\int_0^\infty {\frac{x}{{\left( {x + 1} \right)\sqrt {4{x^4} + 8{x^3} + 12{x^2} + 8x + 1} }}dx}$
- Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves
- Euler type problem, why roots are $m_- < 0 < m_+$
- Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$

For $x<0$, $y”=0$, i.e. $$y=C_1x+C_0.$$

For $x\ge0$, $y”-\frac1by=1$.

Assuming $b>0$, the solution of the homogenous equation is

$$y=A\exp(\frac x{\sqrt b})+B\exp(-\frac x{\sqrt b}),$$

and a particular solution is

$$y=-b.$$

Hence the general solution,

$$y=A\exp(\frac x{\sqrt b})+B\exp(-\frac x{\sqrt b})-b.$$

You will ensure continuity of the function by equating the function and first derivative at $x=0$:

$$C_0=A+B-b\\C_1=\frac A{\sqrt b}-\frac B{\sqrt b}.$$

In particular, starting from the steady state, $A=B=\frac b2$, and

$$y=b\left(\cosh(\frac x{\sqrt b})-1\right)H(x).$$

I’m going to answer the updated question, which asks for the solution of the problem:

$$ \color{green}{y”(x) – \frac{H(x)}{b} y(x) = H(-x) \, c } \tag{1}$$

which can be rewritten equivalently as follows:

$$ \begin{array}{ll}

y_1” – \frac{1}{b} y_1 = 0 & \quad x > 0 \\ \tag{2}

y_2” = c & \quad x < 0

\end{array}$$

where I have denoted $y_1(x)$ as the solution for $x>0$ and $y_2(x)$ the solution for $x<0$. If we integrate both equations we will arrive at:

$$ \color{blue}{y_1(x) = A e^{r_1 x} + B e^{r_2 x}, \quad y_2(x) = \frac{c x^2}{2} + Dx +E} \tag{3}$$

where $r_i$ are the solutions of the characteristic equation $s^2 – 1/b = 0$ and $A, B, D$ and $E$ are constants of integration. To enforce continuity of both the function and its first derivative, we must provide:

$$y_1(0) = y_2(0), \quad y’_1(0) = y_2′(0), \tag{4}$$

which yields to:

\begin{align}

A+B & = E \\

Ar_1 + B r_2 & = D, \tag{5}

\end{align} which gives us the solution, provided $r_1 \neq r_2$:

$$\color{blue}{A = \frac{D-r_2 E}{r_1 – r_2} , \quad B = \frac{D – r_1 E}{r_2 – r_1}} \tag{6}$$

The constants $D$ and $E$ remain unkown until any boundary conditions or initial conditions are specified.

Hope this helps!

Cheers.

I am not sure wether this answer is correct or not, but as far as the step function is only a sign, I brought it out of the integration.

Please note that, for finding the constants, you need the boundary values and compare the value of the function at the point $x=0$.

$$\begin{array}{l}\frac{{{d^2}y}}{{d{x^2}}} – \frac{{H(x)y}}{b} = H(x)\\\frac{{{d^2}y}}{{d{x^2}}} = \frac{{H(x)y}}{b} + H(x)\\\frac{{{d^2}y}}{{d{x^2}}} = \left( {\frac{y}{b} + 1} \right)H(x)\\\frac{{{d^2}y}}{{\left( {\frac{y}{b} + 1} \right)}} = H(x){d^2}x\\\int {\frac{b}{{y + b}}{d^2}y} = H(x)\int {{d^2}x} \\\int {\left\{ {bLn\left( {y + b} \right) + {c_1}} \right\}dy} = H(x)\int {x + {c_3}dx} \\b\left\{ {\left( {y + b} \right)Ln\left( {y + b} \right) – \left( {y + b} \right)} \right\} + {c_1}y + {c_2} = H(x)\left\{ {\frac{1}{2}{x^2} + {c_3}x + {c_4}} \right\}\end{array}$$

- Please explain the logic behind $d(xy) = y(dx) + x(dy)$
- Steinhaus theorem in topological groups
- Evaluate $\lim\limits_{y\to 0}\log(1+y)/y$ without LHR or Taylor series
- The last few digits of $0^0$ are $\ldots0000000001$ (according to WolframAlpha).
- Riemann rearrangement theorem
- Showing a uniformity is complete.
- can one derive the $n^{th}$ term for the series, $u_{n+1}=2u_{n}+1$,$u_{0}=0$, $n$ is a non-negative integer
- Common tangent to two circles
- Is it possible to make the set $M:=\{(x,y)\in\mathbb{R}^2:y=|x|\}$ into a differentiable manifold?
- Prove that a connected space cannot have more than one dispersion points.
- The minimum value of $|z+1|+|z-1|+|z-i|$ for $z \in \mathbb C?$
- Rounding Percentages
- what is this property called?(Field theory)
- Running in the rain, good or bad idea?
- Numbers to the Power of Zero