# Solve second order diff eq with substitution

• Asphyxiated
In summary, the problem involves solving xy''+2y'=12x^{2}, with the given substitution u=y'. The integrating factor is I(x)=e^{\int \frac {2}{x} dx}, and the solution is y=\int 3x^{2}+\frac{C_{1}}{x^{2}}dx = x^{3}-\frac{C_{1}}{x}+C_{2}. The chain rule is used in the process, but it is not necessary to solve the problem.
Asphyxiated

## Homework Statement

Solve:

$$xy''+2y'=12x^{2}$$

with

$$u=y'$$

## Homework Equations

if you have:

$$y'+P(x)y=Q(x)$$

$$I(x)=e^{\int P(x) dx}$$

## The Attempt at a Solution

The only reason I was able to solve this is because I stumbled upon a similar post here but instead of pulling up a thread from over a year ago i decided to post my specific question.

So if:

$$xy''+2y'=12x^{2} \hookrightarrow y''+\frac{2y'}{x}=12x$$

and

$$y'=u \hookrightarrow y''=u\frac{du}{dy}$$

I do not understand why y'' is equal to this. The way I read y'' is d2y/dx2 so why is it u*du/dy as the derivative on the right side and not d/dx? And! why is the derivative of u with respect to y u*du/dy and not just du/dy.

Maybe I have just forgotten something simple from differential calculus but I can't make sense of it on my own. I am not claiming that what is written in TeX above is wrong because the problem seems to work out, I just don't know why that is the way to do it.

Anyway, if we make the substitution:

$$u\frac{du}{dy}+\frac{2u}{x}=12x$$

then

$$I(x)=e^{\int \frac {2}{x} dx} = e^{2 ln|x|}=x^{2}$$

then multiply both sides by I(x):

$$x^{2}u\frac{du}{dy}+2xu=12x^{3}$$

which is:

$$\frac {d}{dx} (ux^{2})=12x^{3}$$

$$ux^{2}=\int 12x^{3} dx =3x^{4}+C_{1}$$

$$u=3x^{2}+ \frac{C_{1}}{x^{2}}$$

$$\frac {dy}{dx}=3x^{2} + \frac {C_{1}}{x^{2}}$$

$$y= \int 3x^{2}+\frac{C_{1}}{x^{2}}dx = x^{3}-\frac{C_{1}}{x}+C_{2}$$

if that is right please tell me why. I just followed by example from someone elses work.

Asphyxiated said:
$$y'=u \hookrightarrow y''=u\frac{du}{dy}$$

I do not understand why y'' is equal to this.
That's just the chain rule.
$$y'' = \frac{du}{dx} = \frac{du}{dy}\frac{dy}{dx} = u\frac{du}{dy}$$
I'm not sure why you'd do it that way, though. You could simply write
$$x u' + 2 u = 12x^2$$
and solve it with the same integrating factor without the unnecessary complication of invoking the chain rule.

oh haha alright. I guess I was confused by the substitution of y'=u again for some reason.
Thanks for the reminder on that!

So I assume this is a correct solution? It's an even problem so I can't look it up.

I didn't see anything obviously wrong, but I didn't look that closely either. You can always check your answer by plugging it back into the original differential equation to see if it's satisfied.

## 1. What is the substitution method for solving second order differential equations?

The substitution method is a technique used to solve second order differential equations by substituting a new variable for the dependent variable. This new variable is chosen in a way that simplifies the equation and makes it easier to solve.

## 2. When should the substitution method be used to solve second order differential equations?

The substitution method should be used when the given differential equation is in the form of x''(t) + p(t)x'(t) + q(t)x(t) = g(t), where p(t) and q(t) are functions of t and g(t) is a function of t. This method is useful for equations that cannot be solved by other techniques, such as the method of undetermined coefficients or variation of parameters.

## 3. How does the substitution method work?

The substitution method works by substituting a new variable, usually denoted as v, for the dependent variable x. This new variable is chosen in a way that cancels out the derivative terms in the equation and simplifies it to a first order differential equation. Once the new equation is solved, the original dependent variable can be substituted back in to find the solution to the original equation.

## 4. What are the steps for using the substitution method to solve a second order differential equation?

The steps for using the substitution method are as follows:

• 1. Identify the dependent variable x and the independent variable t.
• 2. Substitute a new variable v for x, and rewrite the equation in terms of v and t.
• 3. Choose a specific form for v, such as v = x or v = x'.
• 4. Calculate the derivatives of v with respect to t.
• 5. Substitute the derivatives of v into the original equation.
• 6. Solve the resulting first order differential equation for v.
• 7. Substitute back in the original variable x to find the solution to the original equation.

## 5. Are there any limitations to using the substitution method for solving second order differential equations?

Yes, there are some limitations to using the substitution method. It may not work for all types of second order differential equations, and it can be more time-consuming compared to other methods. Additionally, the chosen substitution may not always result in a simpler equation, making it harder to solve. It is important to consider other techniques, such as the method of undetermined coefficients or variation of parameters, if the substitution method does not seem to be effective for a particular equation.

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