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title: Chapter 1. Differential Equations

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# Chapter 1. Differential Equations

## First Order Separable Equations

Can be written in the form $M(x)-N(y)y'=0$, or equivalently, $M(x)dx=N(y)dy$.

When we write the d.e. in this form, we say that we have separated the variables (to 2 sides).

We can solve such a d.e. by integrating w.r.t. x:
$$
\int M(x)dx=\int N(y)dy+c
$$

### Reduction to Separable Form

Certain first order d.e. are not separable but can be made separable by a simple change of variable. This holds for equations of the form
$$
y'=g( {y\over x})
$$
where $g$ is any function of $y\over x$. We set ${y\over x} = v$, then $y=vx$ and $y'=v+xv'$. Thus $v+xv'=g(v)$, which is separable (${dv\over g(v)-v}=g(v)$). We can now solve for $v$, hence obtain $y$.

### Linear Change of Variable

A d.e. of the form $y'=f(ax+by+c)$ where $f$ is continuous and $b\ne 0$ (separable if $b=0$) can be solved by setting $u=ax+by+c$.

## First Order Linear ODEs

$$
{dy\over dx}+P(x)y=Q(x)
$$

where $P$ and $Q$ are functions of $x$, is called a linear first order d.e.

To solve it, define a new function $R(x)$ by $R(x)=e^{\int^xP(s)ds}$ (note that $R'=RP$) so $(Ry)'=RPy+Ry'$. Hence if we multiply both sides of the equation by $R$ we get
$$
Ry'+RPy=RQ
$$
or
$$
(Ry)'=RQ
$$

Then integrate both sides and divide by $R$ to obtain $y$. The function $R$ is called the **INTEGRATING FACTOR** for this equation.

### Reduction to Linear Form

Bernoulli equations of the form $y'+p(x)y=q(x)y^n$ where $n$ is any real number (linear if $n=0$ or $1$). To solve, rewrite it as $y^{-n}y'+y^{1-n}p(x)=q(x)$, and set $y^{1-n}=z$. Then $(1-n)y^{-n}y'=z'$, and the given d.e. becomes $z'+(1-n)p(x)z=(1-n)q(x)$, which is a first order linear d.e.

## Second Order Linear DEs

$$
y''+p(x)y'+q(x)y=F(x)
$$

If $F(x)\equiv0$, the linear d.e. is called *homogeneous*, otherwise *nonhomogeneous*.

### Homogeneous DEs

#### Theorem:

For a **homogeneous** **linear** d.e. $y''+p(x)y'+q(x)y=0$, any linear combination of two solutions on an open interval $I$ is also a solution on $I$. In particular for such an equation, sums and constant multiples of solutions are again solutions.

#### General Solution of Homogeneous Linear Second Order DE

A *general solution* of $y''+py'+qy=0$ on an open interval $I$ is $y=c_1y_1+c_2y_2$, where $y_1$ and $y_2$ are linearly independent solutions of the d.e. and $c_1$, $c_2$ are arbitrary constants.

A *particular solution* of the d.e. on $I$ is obtained if specific values are assigned to $c_1$ and $c_2$.

#### Homogeneous DE with Constant Coefficients

Consider $y''+ay'+by=0$ ($a$, $b$ are constants). $\lambda^2+a\lambda+b=0$ is called the *characteristic equation* (or *auxiliary equation*) of the d.e. The roots of the characteristic equation are:
$$
\lambda_1=\frac{1}{2}(-a+\sqrt{a^2-4b})
$$

$$
\lambda_2=\frac{1}{2}(-a-\sqrt{a^2-4b})
$$

We obtain $e^{\lambda_1x}$ and $e^{\lambda_2x}$ as solutions of the d.e.

- If $a^2-4b>0$ (two real roots $\lambda_1$ and $\lambda_2$):
  - $e^{\lambda_1x}$ and $e^{\lambda_2x}$ are linearly independent solutions of the d.e. on any interval
  - The corresponding general solution of the d.e. is $y=c_1e^{\lambda_1x}+c_2e^{\lambda_2x}$
- If $a^2-4b=0$ (a real double root $\lambda_1=\lambda_2=-\frac{a}{2}$)):
  - $y_1=e^{-\frac{a}{2}x}$
  - $y_2=xe^{-\frac{a}{2}x}$
  - The corresponding general solution is $y=(c_1+c_2x)e^{-\frac{a}{2}x}$
- If $a^2-4b<0$ (complex conjugate roots $\lambda_1$ and $\lambda_2$):
  - Set $\omega=\sqrt{b-\frac{a^2}{4}}$
  - Then $\lambda_1, \lambda_2 = -\frac{a}{2}\pm i\omega$, where $i^2=-1$
  - $y_1=e^{-\frac{a}{2}x}\cos(\omega x)$
  - $y_2=e^{-\frac{a}{2}x}\sin(\omega x)$
  - The corresponding general solution is $y=c_1y_1+c_2y_2=e^{-\frac{a}{2}x}(c_1\cos(\omega x)+c_2\sin(\omega x))$

#### Nonhomogeneous Equations:

A *general solution* of the non-homogeneous d.e. 
$$
y''+p(x)y'+q(x)y=r(x), r(x)\not\equiv0
$$


is of the form $y(x)=y_h(x)+y_p(x)$, where $y_h(x)=c_1y_1(x)+c_2y_2(x)$ is a general solution of the corresponding homogeneous d.e. $y''+p(x)y'+q(x)y=0$ and $y_p(x)$ is any solution of the non-homogeneous d.e. containing no arbitrary constants.

##### Determination of $y_p(x)$

###### (1)  Method of undetermined coefficients

Applies to equations of the form $y''+ay'+by=r(x)$ where $a$ and $b$ are constants and $r(x)$ is a *polynomial*, *exponential function*, *sine* or *cosine*, or sums or products of such functions.

###### Superposition