verify that each given function is a solution of the differential equation ty

bobbie71G

bobbie71G

Answered question

2021-10-19

verify that each given function is a solution of the differential equation tyy=t2;y=3t+t2

Answer & Explanation

lamusesamuset

lamusesamuset

Skilled2021-10-20Added 93 answers

Step 1
We have differentiated y(t)
y(t)=dydt
y(t)=3+2t
Step 2
Plug the values of y(t) and y(t) in the given differential equation
t(3+2t)(3t+t2)t2
3t+2t23tt2t2
y1(t) has satisfied the equation Thus y1(t) is a solution of given differential equation.
Result
y is a solution of given differential equation

RizerMix

RizerMix

Expert2023-05-14Added 656 answers

Step 1:
First, let's find the derivative of y with respect to t. Using the power rule, we have:
dydt=ddt(3t+t2)=3+2t
Step 2:
Next, we substitute y and dydt into the differential equation:
t(dydt)y=t(3+2t)(3t+t2)=3t+2t23tt2=t2
We observe that the expression simplifies to t2, which is indeed equal to the right-hand side of the differential equation.
Since the left-hand side of the differential equation, tyy, equals the right-hand side, t2, we can conclude that the function y=3t+t2 is indeed a solution of the differential equation tyy=t2.
user_27qwe

user_27qwe

Skilled2023-05-14Added 375 answers

Let's start by substituting y=3t+t2 into the differential equation:
t·dydty=t2
Now, let's calculate the derivative of y with respect to t. The derivative of 3t is 3, and the derivative of t2 is 2t. Applying the power rule of differentiation, we have:
dydt=ddt(3t+t2)=3+2t
Substituting this derivative back into the equation, we get:
t·(3+2t)(3t+t2)=t2
Expanding the terms, we have:
3t+2t23tt2=t2
Simplifying the equation, we find:
t2=t2
Since the left side of the equation is equal to the right side, we can conclude that y=3t+t2 is indeed a solution to the given differential equation tyy=t2.
Therefore, the function y=3t+t2 satisfies the differential equation.
karton

karton

Expert2023-05-14Added 613 answers

Answer:
y=3t+t2
Explanation:
To verify that each given function is a solution of the differential equation tyy=t2, we need to substitute the given function y into the equation and see if it satisfies the equation for all values of t.
Let's start by substituting y=3t+t2 into the differential equation:
t(dydt)(3t+t2)=t2
Now, we need to differentiate y with respect to t to find dydt:
y=3t+t2dydt=3+2t
Substituting this back into the equation:
t·(3+2t)(3t+t2)=t2
Simplifying this equation gives us:
3t+2t23tt2=t2t2=t2
As we can see, both sides of the equation are equal, so the given function y=3t+t2 is indeed a solution of the differential equation tyy=t2.
Therefore, we have verified that y=3t+t2 is a solution of the differential equation.

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