Intereting Posts

A counterexample
How is logistic loss and cross-entropy related?
Proof of Dirac Delta's sifting property
Union of ordinals
What is the largest perfect square that divides $2014^3-2013^3+2012^3-2011^3+\ldots+2^3-1^3$
Show $\sum\limits_{d|n}\phi(d) = n$.
Base and subbase of a topology
A problem on continuity of a function on irrationals for $f(x) = \sum_{r_n \leq x} 1/n^2$
The order of the number of integer pairs satisfying certain arithmetical function relationships
Combining rotation quaternions
Is B(H) a Hilbert space?
Asymptotic behavior of the partial sums $\sum\limits_{k=1}^{n}k^{1/4} $
Concavity of the $n$th root of the volume of $r$-neighborhoods of a set
Show that any abelian transitive subgroup of $S_n$ has order $n$
A problem on sinusoids

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?

- Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?
- Covariant and partial derivative commute?
- How prove this $f(a)\le f(b)$
- How to investigate the $\limsup$, the $\liminf$, the $\sup$, and especially the $\inf$ of the sequence $(\sqrt{|\sin{n}|})_{n=1}^{\infty}$?
- Countability of local maxima on continuous real-valued functions
- Why is it important to study the eigenvalues of the Laplacian?

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

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

- show that: $f$ is injective $\iff$ there exists a $g: Y\rightarrow X$ such that $g \circ f = idX$
- Shift Operator has no “square root”?
- Constructive proof of Euler's formula
- Possible to solve this differential equation?
- Proof of uniform convergence and continuity
- Must a monotone function have a monotone derivative?
- Proof to show function f satisfies Lipschitz condition when derivatives f' exist and are continuous
- `“Variation of Constant”` -method to solve linear DYs?
- Are there any continuous functions from the real line onto the complex plane?
- how do you solve $y''+2y'-3y=0$?

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 $x=0$.

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

- Using the Determinant to verify Linear Independence, Span and Basis
- How would one be able to prove mathematically that $1+1 = 2$?
- General Isomorphism, for all algebraic structures
- Did Albert Einstein contribute to math?
- Compact space which is not sequentially compact
- Cyclotomic polynomial
- Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.
- Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 – 1247\sqrt{3})$
- Sum $\frac{1}{6} + \frac{5}{6\cdot 12} + \frac{5\cdot 8}{6\cdot 12\cdot 18} + \frac{5\cdot 8\cdot 11}{6\cdot 12\cdot 18\cdot 24}+\ldots$
- Cubic diophantine equation
- How to solve system of equations with mod?
- Evaluate $ \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx $
- What's the difference between convolution and crosscorrelation?
- Is there a precise mathematical connection between hypergeometric functions and modular forms
- Which rationals can be written as the sum of two rational squares?