# second order differential equation,with constant terms

#### evinda

##### Well-known member
MHB Site Helper
Hello Given the $$x^{2}y''+axy'+by=0$$,I have to show that with replacing $$x$$ with $$e^{z}$$,it becomes a second order differential equation,with constant terms.
I tried to do this and I got this: $$y''+\frac{a}{e^{z}}y'+\frac{b}{e^{2z}}y=0$$.
But,at this equation the terms aren't constant What else could I do??

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hello Given the $$x^{2}y''+axy'+by=0$$,I have to show that with replacing $$x$$ with $$e^{z}$$,it becomes a second order differential equation,with constant terms.
I tried to do this and I got this: $$y''+\frac{a}{e^{z}}y'+\frac{b}{e^{2z}}y=0$$.
But,at this equation the terms aren't constant What else could I do??
Hi evinda!

Note that $y$ actually means $y(x)$, and $y'$ actually means $$\displaystyle \frac{dy}{dx}$$.
You did not replaces those $x$'s yet.

Suppose we define $Y(z) = y(x(z)) = y(e^z)$.
Then according to the chain rule:
$$Y'(z) = \frac{dY(z)}{dz} = \frac{dy(x(z))}{dz} = \frac{dy(x)}{dx} \frac{dx(z)}{dz} = y'(x) \frac{dx(z)}{dz}$$
Or shorter:
$$Y' = \frac{dY}{dz} = \frac{dy}{dx} \frac{dx}{dz} = y' \frac{dx}{dz}$$
Perhaps you can express your differential equation with Y, Y', and Y''?

Btw, I have moved your thread to the sub forum Differential Equations, since that is the topic at hand.

#### evinda

##### Well-known member
MHB Site Helper
Hi evinda!

Note that $y$ actually means $y(x)$, and $y'$ actually means $$\displaystyle \frac{dy}{dx}$$.
You did not replaces those $x$'s yet.

Suppose we define $Y(z) = y(x(z)) = y(e^z)$.
Then according to the chain rule:
$$Y'(z) = \frac{dY(z)}{dz} = \frac{dy(x(z))}{dz} = \frac{dy(x)}{dx} \frac{dx(z)}{dz} = y'(x) \frac{dx(z)}{dz}$$
Or shorter:
$$Y' = \frac{dY}{dz} = \frac{dy}{dx} \frac{dx}{dz} = y' \frac{dx}{dz}$$
Perhaps you can express your differential equation with Y, Y', and Y''?

Btw, I have moved your thread to the sub forum Differential Equations, since that is the topic at hand.
I found this:
$$x^{2}y''\frac{dx}{dz}+x^{2}y'\frac{d^{2}x}{dz^{2}}+axy'\frac{dx}{dz}+by=0$$
Is this right??If yes,how can I continue?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
I found this:
$$x^{2}y''\frac{dx}{dz}+x^{2}y'\frac{d^{2}x}{dz^{2}}+axy'\frac{dx}{dz}+by=0$$
Is this right??If yes,how can I continue?
Looks like you substituted it the wrong way around.

From:
$$Y' = y' \frac{dx}{dz}$$
we get:
$$y' = \frac{Y'}{\frac{dx}{dz}} = \frac{Y'}{e^z} = \frac{Y'}{x}$$
Perhaps you can substitute that in the original DE?

#### evinda

##### Well-known member
MHB Site Helper
Looks like you substituted it the wrong way around.

From:
$$Y' = y' \frac{dx}{dz}$$
we get:
$$y' = \frac{Y'}{\frac{dx}{dz}} = \frac{Y'}{e^z} = \frac{Y'}{x}$$
Perhaps you can substitute that in the original DE?
So,is it like that:

$$y''=\frac{y''(z)}{\frac{dx}{dz}}-y'(z)$$ ?

Staff member

#### chisigma

##### Well-known member
Hello Given the $$x^{2}y''+axy'+by=0$$,I have to show that with replacing $$x$$ with $$e^{z}$$,it becomes a second order differential equation,with constant terms.
I tried to do this and I got this: $$y''+\frac{a}{e^{z}}y'+\frac{b}{e^{2z}}y=0$$.
But,at this equation the terms aren't constant What else could I do??
The standard way to transform an ODE of this type [Euler-Cauchy differential equation...] is in the substitution $u = \ln x$, so that You have...

$\displaystyle \frac{d y}{d x} = \frac{d y}{d u} \frac{d u} {d x} = \frac{1}{x} \ \frac{dy}{d u}\ (1)$

$\displaystyle \frac{d^{2} y}{d x^{2}} = \frac{d^{2} y}{d u^{2}} (\frac{d u}{d x})^{2} + \frac{d y}{d u} \frac{d^{2} u}{d x^{2}} = \frac{1}{x^{2}} (\frac{d^{2} y}{d u^{2}} - \frac{d y}{d u})\ (2)$

Inserting (1) and (2) into the original ODE You obtain...

$\displaystyle \frac{d^{2} y}{d u^{2}} + (a -1)\ \frac{d y}{d u} + b\ y =0\ (3)$

A 'very pratical' way to attack this type of equation is however to search solutions of the form $y = x^{\nu}$. Imposing that You arrive at a second order algebraic equation in $\nu$ and if $\nu_{1}$ and $\nu_{2}$ are the solutions, then the general solution of the ODE is...

$\displaystyle y(x) = c_{1}\ x^{\nu_{1}} + c_{2}\ x^{\nu_{2}}\ (4)$

Kind regards

$\chi$ $\sigma$

#### evinda

##### Well-known member
MHB Site Helper
The standard way to transform an ODE of this type [Euler-Cauchy differential equation...] is in the substitution $u = \ln x$, so that You have...

$\displaystyle \frac{d y}{d x} = \frac{d y}{d u} \frac{d u} {d x} = \frac{1}{x} \ \frac{dy}{d u}\ (1)$

$\displaystyle \frac{d^{2} y}{d x^{2}} = \frac{d^{2} y}{d u^{2}} (\frac{d u}{d x})^{2} + \frac{d y}{d u} \frac{d^{2} u}{d x^{2}} = \frac{1}{x^{2}} (\frac{d^{2} y}{d u^{2}} - \frac{d y}{d u})\ (2)$

Inserting (1) and (2) into the original ODE You obtain...

$\displaystyle \frac{d^{2} y}{d u^{2}} + (a -1)\ \frac{d y}{d u} + b\ y =0\ (3)$

A 'very pratical' way to attack this type of equation is however to search solutions of the form $y = x^{\nu}$. Imposing that You arrive at a second order algebraic equation in $\nu$ and if $\nu_{1}$ and $\nu_{2}$ are the solutions, then the general solution of the ODE is...

$\displaystyle y(x) = c_{1}\ x^{\nu_{1}} + c_{2}\ x^{\nu_{2}}\ (4)$

Kind regards

$\chi$ $\sigma$
I understand But how from this equation: $\displaystyle \frac{d^{2} y}{d u^{2}} + (a -1)\ \frac{d y}{d u} + b\ y =0\$ can I get the general solution??
I tried like that: $$r^{2}+(a-1)r+b=0 => d=(a-1)^{2}-4b$$
but I don't know how to continue...

#### chisigma

##### Well-known member
I understand But how from this equation: $\displaystyle \frac{d^{2} y}{d u^{2}} + (a -1)\ \frac{d y}{d u} + b\ y =0\$ can I get the general solution??
I tried like that: $$r^{2}+(a-1)r+b=0 => d=(a-1)^{2}-4b$$
but I don't know how to continue...
The only You have to do is to complete the solution of the second order equation...

$\displaystyle r^{2}+(a-1)r + b = 0 \implies r_{1} = \frac{1 - a - \sqrt{(1-a)^{2} - 4 b}}{2},\ r_{2} = \frac{1 - a + \sqrt{(1-a)^{2} - 4 b}}{2}\ (1)$

... and the solution is given by...

$\displaystyle y = c_{1} e^{r_{1}\ u} + c_{2} e^{r_{2}\ u} = c_{1}\ x^{r_{1}} + c_{2}\ x^{r_{2}}\ (2)$

Kind regards

$\chi$ $\sigma$

#### evinda

##### Well-known member
MHB Site Helper
The only You have to do is to complete the solution of the second order equation...

$\displaystyle r^{2}+(a-1)r + b = 0 \implies r_{1} = \frac{1 - a - \sqrt{(1-a)^{2} - 4 b}}{2},\ r_{2} = \frac{1 - a + \sqrt{(1-a)^{2} - 4 b}}{2}\ (1)$

... and the solution is given by...

$\displaystyle y = c_{1} e^{r_{1}\ u} + c_{2} e^{r_{2}\ u} = c_{1}\ x^{r_{1}} + c_{2}\ x^{r_{2}}\ (2)$

Kind regards

$\chi$ $\sigma$
I got it!!!And if I want to look at the same problem for x<0,what do I have to do???Maybe to set $$x=-e^{u}$$,or am I wrong?

#### chisigma

##### Well-known member
I got it!!!And if I want to look at the same problem for x<0,what do I have to do???Maybe to set $$x=-e^{u}$$,or am I wrong?
That is a very interesting question!... we have seen that the solution is of the type...

$\displaystyle y(x) = c_{1}\ x^{r_{1}} + c_{2}\ x^{r_{2}}\ (1)$

A function like $x^{r}$, where r may be any real [or even complex...] number, in general has in x=0 a singularity called brantch point, and that means that from x=0 several brantches of the function merge. For x<0 the function has several brantches and in general has a real and an imaginary part. An interesting example is the function $x^{\sqrt{2}}$ plotted by 'MonsterWolfram'...

x^(sqrt(2)) from -1 to 1 - Wolfram|Alpha

Kind regards

$\chi$ $\sigma$

#### evinda

##### Well-known member
MHB Site Helper
That is a very interesting question!... we have seen that the solution is of the type...

$\displaystyle y(x) = c_{1}\ x^{r_{1}} + c_{2}\ x^{r_{2}}\ (1)$

A function like $x^{r}$, where r may be any real [or even complex...] number, in general has in x=0 a singularity called brantch point, and that means that from x=0 several brantches of the function merge. For x<0 the function has several brantches and in general has a real and an imaginary part. An interesting example is the function $x^{\sqrt{2}}$ plotted by 'MonsterWolfram'...

x^(sqrt(2)) from -1 to 1 - Wolfram|Alpha

Kind regards

$\chi$ $\sigma$
So,what do I have to do to show that $$x^{2}y''+axy'+by=0$$ with $$x<0$$ becomes a second order differential equation,with constant terms? #### chisigma

##### Well-known member
So,what do I have to do to show that $$x^{2}y''+axy'+by=0$$ with $$x<0$$ becomes a second order differential equation,with constant terms? The solving procedure is valid for any value of $- \infty < x < + \infty$... for x<0 there are only some [minor] problems...

Kind regards

$\chi$ $\sigma$

#### evinda

##### Well-known member
MHB Site Helper
The solving procedure is valid for any value of $- \infty < x < + \infty$... for x<0 there are only some [minor] problems...

Kind regards

$\chi$ $\sigma$
Don't I have to set x to something negative?

#### evinda

##### Well-known member
MHB Site Helper
Don't I have to set x to something negative?
Or can I just set x=-p,p>0??