# substitution's questions - Chinese 1answer

871 substitution questions.

### 1 Solving a differential equation via variable substitution

Consider the following differential equation $$\dot{x}(t) = f(x(t))+g(u(t)), \ \ t\ge 0,$$ where $x(0)\in\mathbb{R}$ and $u(t)$ is an external input. Suppose that the functions $f$ and $g$ are such ...

### Confused about solving PDEs by defining new variables

My course's notes say that for a function $u=u(x,y)$, if we define $\alpha = x+ay$, and $\beta = x+by$, then: $u_x = u_\alpha + u_\beta$, and $u_y = au_\alpha + bu_\beta$. And this can help solve ...

### 1 Integrating with respect to $u$ with $x$ values

1 answers, 32 views integration substitution
I'm trying to work out the following indefinite integral: $\int x^3 \sqrt{1-x^2}$. To solve this, I said let $u=1-x^2$. Then, $\frac{du}{dx}=-2x$, and $du=-2x \cdot dx$, and $dx=\frac{du}{-2x}$. ...

### 2 Computing the integral of a rational function

2 answers, 34 views integration substitution
I need to compute the integral $\int \dfrac{2x}{(x^2+x+1)^2} \cdot dx$. I tried using the integration of a rational function technique, with $\frac{Ax+B}{x^2+x+1}+\frac{Cx+D}{(x^2+x+1)^2}$, but this ...

### 2 Can one use any substitution to find the antiderivative without taking the domain into account?

0 answers, 25 views integration substitution
This perhaps might be a simple question but for some reason I can't seem to figure out a satisfying answer for it. Let's consider the following indefinite integral that does not require a ...

### -1 $(a+b^2)(b+c^2)(c+a^2)≤13+…$

Prove the following inequality for non-negative real numbers $a,b,c$: If $a+b+c=3$ then: $$(a+b^2)(b+c^2)(c+a^2)\le13+abc(1-2abc)\qquad(1)$$ There are two more variants of the same problem. The ...

### 2 How do I solve $\int\frac{\sqrt{x}}{\sqrt{x}-3}\,dx?$

$$\int \frac{\sqrt{x}}{\sqrt{x}-3}dx$$ What is the most dead simple way to do this? My professor showed us a trick for problems like this which I was able to use for the following simple example: ...

### 1 Integral with Substituting $u=e^x$

2 answers, 21 views integration substitution

### -1 Need help with substitution for integration

1 answers, 39 views integration derivatives substitution
I'm trying to go through the solution for a SIS epidemic model, but I'm stuck figuring out the following steps. So we have $i′=r(k−i)i−αi$ And to solve this, they did a substitution $y=i^{−1}$. ...

### -1 Need help with solving equations for SIS model

0 answers, 28 views integration derivatives substitution