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4 years ago

What is the general solution to the ODE

1 answer:

5 0

we are given the ordinary differential equation of dy/dx = 2y+1. The ODE is variable separable which means x and y can be grouped altogether explicitly. In this case, the equation becomesdy / 2y + 1 = dx
1/2 ln (2y + 1) = x + ln Cln (2y + 1) = 2x + ln C2y + 1 = C e 2x y = (C e 2x - 1)/2

The ODE y' + xy = x can be grouped into y' + x(y-1) = 0 'y' / y-1 = -xThis is variable separable

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L :V --> W is a linear transformation. Prove each of the following (a) ker L is a subspace of V. (b) range L is a subspace of

iragen [17]

Answer:

a) Assume that $x,y\in\ker L$ , and $\alpha$ is a scalar (a real or complex number).

First. Let us prove that $\ker L$ is not empty. This is easy because $L(0_V)=0_W$ , by linearity. Here, $0_V$ stands for the zero vector of V, and $0_W$ stands for the zero vector of W.

Second. Let us prove that $\alpha x\in\ker L$ . By linearity

$L(\alpha x) = \alpha L(x)=\alpha 0_W=0_W$ .

Then, $\alpha x\in\ker L$ .

Third. Let us prove that $y+ x\in\ker L$ . Again, by linearity

$L(x+y)=L(x)+L(y) = 0_W + 0_W=0_W$ .

And the statement readily follows.

b) Assume that $u$ and $v$ are in range of $L$ . Then, there exist $x,y\in V$ such that $L(x)=u$ and $L(y)=v$ .

First. Let us prove that range of $L$ is not empty. This is easy because $L(0_V)=0_W$ , by linearity.

Second. Let us prove that $\alpha u$ is on the range of $L$ .

$\alpha u = \alpha L(x) = L(\alpha x) = L(z)$ .

Then, there exist an element $z\in V$ such that $L(z)=\alpha u$ . Thus $\alpha u$ is in the range of $L$ .

Third. Let us prove that $u+v$ is in the range of $L$ .

$u+v = L(x)+L(y) = L(x+y)=L(z)$ .

Then, there exist an element $z\in V$ such that $L(z)= u +v$ . Thus $u +v$ is in the range of $L$ .

Notice that in this second part of the problem we used the linearity in the reverse order, compared with the first part of the exercise.

Step-by-step explanation:

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3 years ago

Evaluate the expression 36 + 5^3

d1i1m1o1n [39]

Answer:

161

Step-by-step explanation:

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3 years ago

Read 2 more answers

How do you factor the expression 20k+50

Aliun [14]

Answer is 10(2k+5)

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Work Shown:

20k+50 = 10*2k + 10*5
20k+50 = 10(2k+5)

Note how we can distribute the 10 back in to check our work
10(2k+5) = 10(2k)+10(5) = 20k+50
so that confirms we have the right answer

Another thing to notice is that 10 is the largest factor that we can pull out of 20k and 50. The value 10 is the GCF (greatest common factor) of 20 and 50.

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4 years ago

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What is the value of x?

LiRa [457]

exterior angle is equal to sum of two opposite interior angle

141= 32 + x

141-32= x= 109° ANS.

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2 years ago

The ideal width of a safety belt strap for a certain automobile is 5 cm. An actual width can vary by at most 0.35 cm. Choose the

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