# 2nd order DE's with constant coefficients.

• Jan 8th 2011, 04:31 PM
hejmh
2nd order DE's with constant coefficients.
1- How can I solve this equation? y''+4y'=0

2-What is the diffrential equation of type y''+ay'+by=0 which has the solution of:
e^(3x) and xe^(3x)

• Jan 8th 2011, 04:32 PM
Ackbeet
What ideas have you had so far?
• Jan 8th 2011, 04:38 PM
dwsmith
Quote:

Originally Posted by hejmh
1- How can I solve this equation? y''+4y'=0

2-What is the diffrential equation of type y''+ay'+by=0 which has the solution of:
e^(3x) and xe^(3x)

$y''+4y'=0\Rightarrow m^2+4m=0$

Do you know how to use this form?
• Jan 8th 2011, 04:49 PM
Prove It
For part 2, you're told $\displaystyle y = e^{3x}$ is a solution.

So $\displaystyle y' = 3e^{3x}$ and $\displaystyle y'' = 9e^{3x}$.

Substitute into the DE to get $\displaystyle 9e^{3x} + 3a\,e^{3x} + b\,e^{3x} = 0$.

Now do the same with the other solution $\displaystyle y = x\,e^{3x}$ to get another equation that involves $\displaystyle a$ and $\displaystyle b$. Then you can solve the two equations simultaneously for $\displaystyle a$ and $\displaystyle b$.
• Jan 8th 2011, 05:00 PM
dwsmith
Quote:

Originally Posted by dwsmith
$y''+4y'=0\Rightarrow m^2+4m=0$

Do you know how to use this form?

From this point, you factor and solve for m.

$m(m+4)=0\Rightarrow m=0,-4$

$y=C_1e^{m_1x}+C_2e^{m_2x}$

$y=C_1e^{0*x}+C_2e^{-4x}=C_1+C_2e^{-4x}$
• Jan 8th 2011, 05:02 PM
hejmh
Yes. thanks
• Jan 9th 2011, 08:19 AM
HallsofIvy
Quote:

Originally Posted by hejmh
1- How can I solve this equation? y''+4y'=0

Here's another way to do this. Since there is no "y" (undifferentiated) let u= y' so the equation becomes u'+ 4u= 0. That is the same as du/dx= -4u which is a separable, first order differential equation. You can write it as du/u= -4 dx and integrate both sides: ln(u)= -4x+ c which leads to $u(x)= y'(x)= e^{-4x+ c}= Ce^{-4x}$ where $C= e^{c}$. Now find y by integrating $y'= Ce^{-4x}$: $y(x)= \frac{C}{-4}e^{4x}+ B$ which we can also write as $y(x)= Ae^{-4x}+ B$ where now $A= \frac{C}{-4}= \frac{e^c}{-4}$.