Second Order Linear ODE

What is the general solution to the DE $\frac{\text{d}^2x}{\text{d}y^2}+7\frac{\text{d}x}{\text{d}y}-8y=0$  ?

•  $y=Ce^{-x}+De^{8x}$  

•  $y=Ce^x+De^{-8x}$  

•  $y=Ce^x+De^{7x}$  

•  $y=Ce^{-x}+De^{7x}$  

 $\frac{\text{d}^2x}{\text{d}y^2}+4\frac{\text{d}x}{\text{d}y}-6y=0$  

Which of the following options are TRUE about the above DE?

• The roots of the auxiliary equations are two complex roots.

• The auxiliary equation has two different roots.

• The auxiliary equation has two equal roots.

• The equation is homogeneous.

Given  $\frac{\text{d}^2x}{\text{d}y^2}+4\frac{\text{d}x}{\text{d}y}+3y=2e^{-x}.$  

The correct  $y_p$  is

•  $Ce^{-x}$  

•  $C\cos x+D\sin x$  

•  $Ce^x$  

•  $Cxe^{-x}$  

 $\frac{\text{d}^2x}{\text{d}y^2}-y=\cot x$  
The above DE can be solved using the undetermined coefficient meThod.

• TRUE

• FALSE

Find a general solution of the following homogeneous linear second order differential equation.

y "- 2 y '- 8 y=0

• y = c₁e^2x + c₂e^4x

• y= c₁e^-2x + c₂e^4x

• y = c₁e^2x + c₂e^-4x

• y = c₁e^-2x + c₂e^-4x

The following non-homogeneous linear second order ODE has y_h=c₁e^x+c₂xe^x . What is its y_p if it is to be solved using Undetermined Coefficient Approach?

y "-2 y '+ y = xe^x

• (1/6)(x³e^x)

• (1/6)(xe^x)

• (1/6)(x²e^x)

• x³e^x

Find the characteristics equation for y"+4 y'+4 y= 2x + 6.

• m²+4m+4=0

• m²+4m+4=2x+6

• m²+4m=0

• m²+4=0

The nonhomogeneous 2nd order differential equation can be solved by

• separating the variables

• undetermined coefficient method

• variation of parameters method

• wronskian method

The initial guess for $f\left(x\right)=4\cos3x+2x^2$  

•  $y_p=\left(C\cos3x+Fx^2\right)$  

•  $y_p=\left(C\cos3x+D\sin3x+Fx^2+Gx+H\right)$  

•  $y_p=\left(C\cos3x+D\sin3x+Fx^2\right)$  

•  $y_p=\left(C\cos3x-D\sin3x+Fx^2+G\right)$  

 $x^2\frac{\text{d}^2x}{\text{d}y^2}-5y=x+1$  is a non homogeneous second order differential equations with constant coefficients.

• TRUE

• FALSE