Free Printable Second Order Linear ODE

1.

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

a)

$y=Ce^{-x}+De^{8x}$

b)

$y=Ce^x+De^{-8x}$

c)

$y=Ce^x+De^{7x}$

d)

$y=Ce^{-x}+De^{7x}$

2.

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

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

a)

The roots of the auxiliary equations are two complex roots.

b)

The auxiliary equation has two different roots.

c)

The auxiliary equation has two equal roots.

d)

The equation is homogeneous.

3.

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

a)

$Ce^{-x}$

b)

$C\cos x+D\sin x$

c)

$Ce^x$

d)

$Cxe^{-x}$

4.

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

a)

TRUE

b)

FALSE

5.

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

y "- 2 y '- 8 y=0

a)

y = c₁e^2x + c₂e^4x

b)

y= c₁e^-2x + c₂e^4x

c)

y = c₁e^2x + c₂e^-4x

d)

y = c₁e^-2x + c₂e^-4x

6.

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

a)

(1/6)(x³e^x)

b)

(1/6)(xe^x)

c)

(1/6)(x²e^x)

d)

x³e^x

7.

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

a)

m²+4m+4=0

b)

m²+4m+4=2x+6

c)

m²+4m=0

d)

m²+4=0

8.

The nonhomogeneous 2nd order differential equation can be solved by

a)

separating the variables

b)

undetermined coefficient method

c)

variation of parameters method

d)

wronskian method

9.

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

a)

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

b)

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

c)

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

d)

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

10.

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

a)

TRUE

b)

FALSE

Second Order Linear ODE

Total questions: 10

Worksheet time: 21mins

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