Question

In each of Problems 1 through 6, determine the order of the given differential equation; alsostate whether the equation is linear or nonlinear.(1) t^2 (d^2y/dt^2)+ t(dy/dt)+ 2y = sin t(2) (1 + y^2)(d^2y/dt^2)+ t(dy/dt)+ y = e^t(3) (d^4y/dt^4)+(d^3y/dt^3)+(d^2y/dt^2)+(dy/dt)+ y =1(4) (dy/dt)+ ty^2 = 0(5) (d^2y/dt^2)+ sin(t + y) = sin t

Answer 1

Answer:

The question is in complete, below is the complete question

"In each of Problems 1 through 6, determine the order of the given differential equation; also state whether the equation is linear or nonlinear

1. $t^{2}\frac{d^{2}y}{dt^{2}} +t\frac{dy}{dt}+2y=sint$

2.$(1+y^{2} )\frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}+y=e^{t}$

3.$\frac{d^{4}y }{dt^{4}}+ \frac{d^{3}y }{dt^{3}}+ \frac{d^{2}y }{dt^{2}}+ \frac{dy}{dt}+y=1$

4. $\frac{dy}{dt}+ty^{2}=0$

5.$\frac{d^{2}y }{dt^{2}}+sin(t+y)=sint$

6.$\frac{d^{3}y }{dt^{3}}+t\frac{dy }{dt}+cos^{2}(t)y=t^{3}$

"

below is the answer

1. second order, linear  

2. second order, non- linear  

3. fourth order, linear

4. first order, non- linear

5.second order, non- linear

6. third order, linear  

Step-by-step explanation:

To determine the order of a differential equation, we look out for the highest derivative in the equation. if the power of the highest derivative is one, we call it a first order  differential equation and if the highest power of the derivative is 2 we call it a second order differential equation.

Also a differential equation is said to be linear if the independent variable and its derivative are linear.

Now let use the above condition to check the following

1.$t^{2}\frac{d^{2}y}{dt^{2}} +t\frac{dy}{dt}+2y=sint$

the highest power of the derivative is 2 and the independent variable and its derivatives are linear hence we conclude second order, linear  

2.$(1+y^{2} )\frac{d^{2}y}{dt^{2}}+t\frac{dy}{dt}+y=e^{t}$

the highest power of the derivative is 2 and the $y^{2}\frac{d^{2}y}{dt^{2}}$ is non-linear hence we conclude second order, non- linear  

3.$\frac{d^{4}y }{dt^{4}}+ \frac{d^{3}y }{dt^{3}}+ \frac{d^{2}y }{dt^{2}}+ \frac{dy}{dt}+y=1$

the highest power of the derivative is 4 and the independent variable and its derivatives are linear hence we conclude fourth order, linear

4. $\frac{dy}{dt}+ty^{2}=0$

the highest power of the derivative is 1 and the $y^{2}$ is non-linear hence we conclude first order, non- linear  

5.$\frac{d^{2}y }{dt^{2}}+sin(t+y)=sint$

the highest power of the derivative is 2 and the $(t+y)$ is non-linear hence we conclude second order, non- linear  

6.$\frac{d^{3}y }{dt^{3}}+t\frac{dy }{dt}+cos^{2}(t)y=t^{3}$

the highest power of the derivative is 3 and the independent variable and its derivatives are linear hence we conclude third order, linear