Find a General Solution to the Given Cauchydasheuler Equation
Question
Find general solution to the given Cauchy-Euler equation for t > 0.28y =The general solution is y(t) =
Find general solution to the given Cauchy-Euler equation for t > 0. 28y = The general solution is y(t) =
Answers
Find the general solution to the differential equation $$\frac{d y}{d t}=2 t$$
We need to find the general solution to the differential equation. Dy I'm T T Z. Do you square? So we start off by separating the variables so let's move tt with teeth and we will integrate both sides. This gives us why is in the integral of he squared is T cube over three. And since we have no limit of integration, we have to add our plus C. We can only find see if they give us initial conditions. But since they did not our answer To the general solution. So the general solution is t cubed over three plus E.
We need to find the general solution to the differential equation. Do you want to buy? DT is equal to eat to the teeth? So we start by separating the variables need to the T. D. Team and then we integrate both sides. We get why is equal to this integral is also E. T plus C. And they are asking for the general solution. So this is the general solution. The particular solution comes when you find see and that is if they give you initial conditions, so you can only find see if they give you some other amounts of information. The general solution is why is equal to E to the T plus C.
Additions this differential equation. Well, we could take the end up in an egg roll. A ball signs with Let's see, we're going toe. He squared over two. How about I know thi my That's integral. Uh, t squared over two times. One of the team Katie. And then we can forget about her. Plus one of pants. We'll have a bus team. Simplify that a bit. What? But see, he is a pencil out. We have t squared over two on a team minus T squared over. Form plus team person. Unknown constancy. Okay, awesome.
So we are given an executive torrential occasion. G I T T is equal to three T squared. Of course T cubed. And we have to find the general solution. Now by integrating both sides with respect to T. We get our Y is equal to the integral three T squared. Of course. Too cute. Two key. Now in many of these questions you have to use recognition of uh common integral of trig functions and um these algebraic terms. So here we know that The three T Squared. If we integrate this we actually get T cube right? Of course. Plus a constant. Let's ignore that for now now. And our if we integrate course tea with respect to T, we actually get signed same to you right now. You will recognize that here there has been applied a chain rule too sign T cubed. And if we try to differentiate this we actually get our team here. Therefore you buy recognition. You see that are Y is equal to sign T. Cute and adding the constant of integration. We get see here. So this is our solution and final answer.
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Find a General Solution to the Given Cauchydasheuler Equation
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