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First order differential equations

Bat

New member
Oct 25, 2021
4
Hi,
Is the answer:
y(x) _homogenous =v(x)
y(x) _private =u(x)v(x)
?
Or they refer to something else?
I dont know how to approach to it
IMG_20211114_185403.jpg
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
881
The given equation is \(\displaystyle \frac{dy}{dx}+ p(x)y(x)= q(x)\). The "associated homogeneous equation" is \(\displaystyle \frac{dy}{dx}+ p(x)y(x)= 0\). (As they say, "q(x)= 0").

That equation is "separable". \(\displaystyle \frac{dy}{dx}= -p(x)y(x)\) and then \(\displaystyle \frac{dy}{y}= -p(x)dx\).

Integrating both sides the general solution to the associated homogeneous equation is \(\displaystyle ln(y(x))= -\int p(x)dx+ c\).
Take the exponential of both sides- \(\displaystyle y(x)= e^{-\int p(x)dx+ c}= Ce^{-\int p(x)dx}\) where \(\displaystyle C= e^c\).

Now, they are saying that you should look for a solution of the form "y(x)= v(x)+ u(x)v(x) where v(x) is a solution of the homogenous equation" (I would call this method "variation of parameters" rather than "variable parameters" also, since v(x) satisfies the homogenous equation, putting it alone into the equation will give 0 so I would use just y(x)= u(x)v(x). ) so \(\displaystyle y(x)= u(x)e^{-\int p(t)dt}\)).

Then \(\displaystyle y'= u'e^{-\int p(x)dx}- u(x)\left(p(x)e^{-\int p(x)dx}\right)= u'v- p(x)u(x)v\) and \(\displaystyle y'+ p(x)y= u'v- p(x)u(x)v+ p(x)u(x)v= u'v= q(x)\). So \(\displaystyle u'= \frac{du}{dx}= \frac{q(x)}{v(x)}=\)\(\displaystyle \frac{q(x)}{e^{-\int p(t)dt}}\) and \(\displaystyle u(x)= \int \frac{q(x)}{e^{-\int p(t)dt}}dx\).

For a very simple example, consider \(\displaystyle y'- 2y= x^2\). p(x) is the constant -2 and q(x) is \(\displaystyle x^2\).

The associated homogeneous equation is \(\displaystyle y'- 2y= 0\) or \(\displaystyle y'= \frac{dy}{dx}= 2y\) which can be separated as \(\displaystyle \frac{dy}{y}= 2dx\). Integrating \(\displaystyle ln(y)= 2x+ c\) and, solving for y, \(\displaystyle y(x)= Ce^{2x}\).

Now, using "variational constants" we look for a solution to the entire equation of the form \(\displaystyle y(x)= u(x)e^{2x}\). Then \(\displaystyle y'= u'e^{2x}+ 2ue^{2x}\) so the equation becomes \(\displaystyle y'- 2y= u'e^{2x}+ 2ue^{2x}- 2ue^{2x}= u'e^{2x}= x^2[/tex\).

\(\displaystyle u'= \frac{du}{dx}= x^2e^{-2x}\) so \(\displaystyle u= \int x^2 e^{-2x}dx\).

To integrate that, use "integration by parts, taking \(\displaystyle u= x^2\), so that \(\displaystyle du= 2xdx\), and \(\displaystyle dv= e^{-2x}dx\) so that \(\displaystyle v= -\frac{1}{2}e^{-2x}\).

\(\displaystyle \int x^2e^{-2x}dx= -\frac{1}{2}x^2e^{-2x}+ \int xe^{-2x}dx\).

To integrate \(\displaystyle \int xe^{-2x}dx\) use "integration by parts" again this time with \(\displaystyle u= x\) so that \(\displaystyle du= dx\)and \(\displaystyle dv= e^{-2x}dx\) so that \(\displaystyle v= -\frac{1}{2}e^{-2x}\).

\(\displaystyle \int xe^{-2x}dx= -\frac{1}{2}xe^{-2x}+ \frac{1}{2}\int e^{-2x}dx= -\frac{1}{2}xe^{-2x}- \frac{1}{4}e^{-2x}+ C\).