Chapter3Section3

Chapter3Section3 - 3.3 Homogeneous Equations with Constant...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3.3: Homogeneous Equations with Constant Coefficients Remark Assume that y H x L = e r x is a solution to the n th order, linear, homogeneous differential equation a n y H n L + ” + a y = 0. Then, since d i dx i e rx = r i e rx , for 1 £ i £ n , we have a n r n e r x + a n- 1 r n- 1 e r x + ” + a e r x = if and only if a n r n + a n- 1 r n- 1 + ” + a = 0. Definition The characteristic equation for the n th order, linear, homogeneous differential equation a n y H n L + ” + a y = 0. is given by a n r n + a n- 1 r n- 1 + ” + a = 0. Theorem - Solutions to the n th Order, Linear, Homogeneous DE with Constant Coefficients: Distinct and Real If the roots r 1 , ..., r n of the characteristic equation are real and distinct, Then y H x L = c 1 e r 1 x + c 2 e r 2 x + ” + c n e r n x is a general solution to a n y H n L + ” + a y = 0. Example Solve the IVP y I 3 M + 2 y ''- 5 y '- 6 y = 0, y H L = 1, y ' H L = 2, y '' H L = 3. Mathematica Example Solve the IVP y I 3 M + 2 y ''- 5 y '- 6 y = 0, y H L = 1, y ' H L = 2, y '' H L = 3 using Mathematica . First, let's factor the characteristic equation... In[5]:= Factor @ r^3 + 2 r^2- 5 r- 6 D Out[5]= H- 2 + r L H 1 + r L H 3 + r L Or we could just solve... In[6]:= Solve @ r^3 + 2 r^2- 5 r- 6 0, r D Out[6]= 88 r fi - 3 < , 8 r fi - 1 < , 8 r fi 2 << Hence, using our theory, we see that the general solution is given by...
View Full Document

{[ snackBarMessage ]}

Page1 / 10

Chapter3Section3 - 3.3 Homogeneous Equations with Constant...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online