MA1035: Engineering Modeling using Dynamic Systems

Session 03: Higher Order Linear Differential Equations with cc

Prof. Alejandro Ucan-Puc

Tecnológico de Monterrey • Departamento de Ciencias

Goals

  • Describe the general form of a linear homogeneous differential equation with constant coefficients and the IVP.

  • Determine the general solution for a linear homogeneous differential equations with constant coefficients.

  • Deduce the method to solve linear homogeneous differential equations with constant coefficients.

  • Apply the method to solve linear homogeneous differential equations with constant coefficients.

Linear homogeneous differential equations with constant coefficients.

Definition: the generic form of a linear homogeneous differential equation of order $n$ is: $$a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdots+a_2(x)y''+a_1(x)y'+a_0(x)y=g(x).$$

To define an IVP we need to add the initial conditions: $$y(x_0)=y_0,\, y'(x_0)=y_1,\,y''(x_0)=y_2,\cdots,\,y^{(n-1)}(x_0)=y_{n-1}.$$

Example 1:

For the $3y'''+7y''-7y'-3y=0$ we have that $y(x)=ce^{x}$ and $y(x)=ce^{-x/3}+ke^{-3x}$ are solutions.



The function $y=3e^{2x}+e^{-2x}-3x$ is the solution for the IVP $y''-4y=12x$ with initial conditions $y(0)=4$ and $y'(0)=1.$

First Order LODE with cc.

The general form of a first order LODE with cc is $ay'+by=g(x),$ lets try to solve the homogeneous case.

  • The solution have to be of the form $y=e^{mx}.$

  • Substituting in the LODE we have that $$ame^{mx}+be^{mx}=0=e^{mx}(am+b),$$
  • From the fact that $e^{mx}\neq 0$ we have that $$am+b=0\Rightarrow m=-b/a.$$
  • Therefore the solution is $y(x)=ce^{-bx/a}.$

Second Order LODE with cc.

Lets take the second order homogeneous LODE with cc.

  • The solution have to be of the form $y=e^{mx}.$

  • Substituting in the LODE we obtain that our auxiliary polynomial is $$am^2+bm+c=0.$$

  • We have that there two roots to this polynomial (they might be egual): $m_1$ and $m_2.$

  • Remember that if:
  • If $\Delta=b^2-4ac>0$ then the roots are real and different.
  • If $\Delta=0$ then the roots are real and equal.
  • If $\Delta<0$ then the roots are complex conjugates.

First Case: Different real roots:

In the case the roots are real and different, $m_1$ and $m_2,$ then the general solution is $$y=c_1 e^{m_1 x}+c_2e^{m_2 x}.$$


Example 1:

Find the general solution to $2y''-5y'-3y=0,$


Solution:

  • The auxiliary polynomial is $2m^2-5m-3=0$

  • The roots are $m_1=3$ and $m_2=-1/2$

  • Therefore, the general solution is: $$y=c_1e^{3x}+c_2e^{-x/2}.$$

Second Case: Equal real roots:

In the case the roots are real and equal, $m_1=m_2=m,$ then the general solution is $$y=c_1 e^{mx}+c_2xe^{mx}.$$


Example 2:

Find the general solution to $y''-10y'+25y=0.$


Solution:

  • The auxiliary polynomial is $m^2-10m+25=0$

  • The roots are $m_1=m_2=5$

  • Therefore, the general solution is: $$y=c_1e^{5x}+c_2xe^{5x}.$$

Third Case: Complex conjugate roots:

In the case the roots are complex conjugates, $m_1=a+bi$ and $m_2=a-bi,$ then the general solution is $$y=c_1 e^{ax}\cos(bx)+c_2e^{ax}\sin(bx).$$


Example 3:

Find the general solution to $y''+4y'+7y=0.$


Solution:

  • The auxiliary polynomial is $m^2+4m+7=0$

  • The roots are $m_1=-2+\sqrt{3}i$ and $m_2=-2-\sqrt{3}i.$

  • Therefore, the general solution is: $$y=c_1e^{-2x}\cos(\sqrt{3}x)+c_2e^{-2x}\sin(\sqrt{3}x).$$

Example 4:

Find the solution of $$4y''+4y'+17y=0,$$ given $y(0)=-1$ and $y'(0)=2.$

Solution:

  • Auxiliary polynomial: $4m^2+4m+17=0$, with roots $m=-\frac{1}{2}\pm2i$.
  • General solution: $$y=e^{-x/2}\left(c_1\cos(2x)+c_2\sin(2x)\right).$$

Example 4: Applying the Initial Conditions

From $y(0)=-1$, we get $c_1=-1$. Differentiating the general solution and applying $y'(0)=2$ gives

$$-\frac{1}{2}c_1+2c_2=2\quad\Longrightarrow\quad c_2=\frac{3}{4}.$$

Therefore, the solution is

$$y=e^{-x/2}\left(-\cos(2x)+\frac{3}{4}\sin(2x)\right).$$

What to do for order $\geq 2$?

1. Find the auxiliary polynomial.

2. Find the roots of the auxiliary polynomial.

3. Write the general solution using the roots, as follows:
* For every $m_i$ real root, with no repetition; add a tem $c_{i}e^{m_ix}.$
* For every $m_i$ real root, with multiplicity $k,$ add a term $$c_{i1}e^{m_ix}+c_{i2}xe^{m_ix}+\cdots+c_{ik}x^{k-1}e^{m_ix}$$ as many terms as the root multiplcity.
* For every pair of complex conjugate roots $m_i=a+bi$ and $m_{i+1}=a-bi,$ add a term $$e^{ax}(c_{i1}\cos(bx)+c_{i2}\sin(bx)).$$

Summary & Next Step

Homogeneous linear differential equations with constant coefficients, characteristic auxiliary equations, real distinct, repeated, and complex roots.

Next Session: Non-homogeneous Higher Order LODEs with cc