MATH34032_Video2followup

In general, how many solutions are there for a linear second order ODE?

• Two solutions.

• One solution.

• Infinitely many solutions.

• No solutions.

The general solution of a inhomogeneous linear second order ODE takes the form

 $u\left(x\right)=c_1u_1\left(x\right)+c_2u_2\left(x\right)+u_p\left(x\right).$ The first part of this solution,  $c_1u_1\left(x\right)+c_1u_2\left(x\right),$  is ... 

• ... a particular solution.

• ... a general solution of the corresponding homogeneous ODE.

• ... the complementarity solution.

• ... always of the form  $c_1e^{m_+x}+c_2e^{m_-x}.$  

What is a complementary solution for the ODE $u''-2u'-3u=1?$  

•  $u_c\left(x\right)=c_1e^{3x}+c_2e^{-x}$  

•  $u_c\left(x\right)=c_1e^x+c_2e^{-3x}$  

•  $u_c\left(x\right)=-\frac{1}{3}$  

•  $u_c\left(x\right)=c_1x+c_2x^{-3}$  

Which of the following are general solutions of the ODE  $u''-10u'+25u=0?$ (Choose all options that are correct.)

•  $u\left(x\right)=c_1\left(x+1\right)e^{5x}+c_2\left(x-1\right)e^{5x}$  

•  $u\left(x\right)=c_1e^{5x}+c_2xe^{5x}$  

•  $u\left(x\right)=c_1e^{5x}+c_2e^{25x}$  

•  $u\left(x\right)=c_1e^{5x}+c_2\ln\left(x\right)e^{5x}$  

Find a general solution for the ODE $x^2u''+2xu'-2u=0$  on the domain  $x>0$ .

•  $u\left(x\right)=c_1x+c_2x^{-2}$  

•  $u\left(x\right)=c_1x^2+c_2x^{-1}$  

•  $u\left(x\right)=c_1x^{-2}+c_2\ln\left(x\right)x^{-2}$  

•  $u\left(x\right)=c_1e^x+c_2e^{-2x}$