Author Topic: Q1-TUT0501  (Read 2355 times)

Mariah Stewart

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Q1-TUT0501
« on: January 25, 2018, 04:51:29 PM »
Find the general solution of the given differential equation, and use it to determine how solutions as t approaches infinity.

ty' + 2y = sin(t), t>0

Answer: Find an integrating factor


Mariah Stewart

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Re: Q1-TUT0501
« Reply #1 on: January 25, 2018, 04:52:52 PM »
Here's a clearer image

Zihan Wan

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TUT 0501
« Reply #2 on: January 25, 2018, 04:54:59 PM »
Find the general solution of the given differential equation, and use it to determine how solutions behave as t→∞
ty′+2y=sin(t), t>0

Meng Wu

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Re: Q1-TUT0501
« Reply #3 on: January 25, 2018, 06:22:17 PM »
$$ty'+2y=sint$$
First, we divide both sides of the given equation by $t$, we get: $\\$
$$y'+{2\over t}y={sint\over t}$$
Now the differential equation has the form
$$y'+p(t)y=g(t)$$
Hence $p(t)={2\over t}$ and $g(t)={sint\over t}$$\\$
First, we find the integrating factor $\mu(t)$ $\\$
As we know. $\mu(t)=\exp^{\int{p(t)dt}}$ $\\$
Thus, $\mu(t)=\exp^{\int{{2\over t}dt}}=e^{2ln|t|}=e^{ln|t|}\cdot e^{ln|t|}=t\cdot t=t^{2}$$\\$
Then mutiply $\mu(t)$ to both sides of the equation, we get:
$$t^2y'+2ty=tsint$$
and $$(t^2y)'=tsint$$
Integrating both sides:
$$\int{(t^2y)'}=\int{tsint}$$
Thus, $$t^2y=\int{tsint}$$
For $\int{tsin(t)}$, we use Integration By Parts:$\\$
Let $u=t, dv=sint$.$\\$
Then $du=dt, v=-cost$$\\$
Hence, $$\int{tsint}=uv-\int{vdu}$$
$$\int{tsint}=-tcost-\int{-costdt}$$
$$\int{tsint}=-tcost+\int{costdt}$$
$$\int{tsint}=-tcost+sint+c$$
Thus $$t^2y=-tcost+sint+c$$
where $c$ is arbitrary constant.$\\$
Now we divide both sides by $t^2$, we get the general solution:
$$y={(sint-tcost+c)/t^2}$$
Since given $t>0$, $y\rightarrow 0$ as $t \rightarrow \infty$.

Meng Wu

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Re: TUT 0501
« Reply #4 on: January 25, 2018, 06:22:29 PM »
$$ty'+2y=sint$$
First, we divide both sides of the given equation by $t$, we get: $\\$
$$y'+{2\over t}y={sint\over t}$$
Now the differential equation has the form
$$y'+p(t)y=g(t)$$
Hence $p(t)={2\over t}$ and $g(t)={sint\over t}$$\\$
First, we find the integrating factor $\mu(t)$ $\\$
As we know. $\mu(t)=\exp^{\int{p(t)dt}}$ $\\$
Thus, $\mu(t)=\exp^{\int{{2\over t}dt}}=e^{2ln|t|}=e^{ln|t|}\cdot e^{ln|t|}=t\cdot t=t^{2}$$\\$
Then mutiply $\mu(t)$ to both sides of the equation, we get:
$$t^2y'+2ty=tsint$$
and $$(t^2y)'=tsint$$
Integrating both sides:
$$\int{(t^2y)'}=\int{tsint}$$
Thus, $$t^2y=\int{tsint}$$
For $\int{tsin(t)}$, we use Integration By Parts:$\\$
Let $u=t, dv=sint$.$\\$
Then $du=dt, v=-cost$$\\$
Hence, $$\int{tsint}=uv-\int{vdu}$$
$$\int{tsint}=-tcost-\int{-costdt}$$
$$\int{tsint}=-tcost+\int{costdt}$$
$$\int{tsint}=-tcost+sint+c$$
Thus $$t^2y=-tcost+sint+c$$
where $c$ is arbitrary constant.$\\$
Now we divide both sides by $t^2$, we get the general solution:
$$y={(sint-tcost+c)/t^2}$$
Since given $t>0$, $y\rightarrow 0$ as $t \rightarrow \infty$.

Victor Ivrii

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Re: TUT 0501
« Reply #5 on: February 02, 2018, 07:47:18 AM »
ZihanWan Please, no pdf attachments...