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2 years ago

PLS HELP ASAP!!!!!! THANK UUU

Mathematics

1 answer:

Elodia [21]2 years ago

8 0

Answer:

a: is 36

Step-by-step explanation:

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(X^2+y^2+x)dx+xydy=0 Solve for general solution

aksik [14]

Check if the equation is exact, which happens for ODEs of the form

$M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0$

if $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ .

We have

$M(x,y)=x^2+y^2+x\implies\dfrac{\partial M}{\partial y}=2y$

$N(x,y)=xy\implies\dfrac{\partial N}{\partial x}=y$

so the ODE is not quite exact, but we can find an integrating factor $\mu(x,y)$ so that

$\mu(x,y)M(x,y)\,\mathrm dx+\mu(x,y)N(x,y)\,\mathrm dy=0$

is exact, which would require

$\dfrac{\partial(\mu M)}{\partial y}=\dfrac{\partial(\mu N)}{\partial x}\implies \dfrac{\partial\mu}{\partial y}M+\mu\dfrac{\partial M}{\partial y}=\dfrac{\partial\mu}{\partial x}N+\mu\dfrac{\partial N}{\partial x}$

$\implies\mu\left(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}\right)=M\dfrac{\partial\mu}{\partial y}-N\dfrac{\partial\mu}{\partial x}$

Notice that

$\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}=y-2y=-y$

is independent of x, and dividing this by $N(x,y)=xy$ gives an expression independent of y. If we assume $\mu=\mu(x)$ is a function of x alone, then $\frac{\partial\mu}{\partial y}=0$ , and the partial differential equation above gives

$-\mu y=-xy\dfrac{\mathrm d\mu}{\mathrm dx}$

which is separable and we can solve for $\mu$ easily.

$-\mu=-x\dfrac{\mathrm d\mu}{\mathrm dx}$

$\dfrac{\mathrm d\mu}\mu=\dfrac{\mathrm dx}x$

$\ln|\mu|=\ln|x|$

$\implies \mu=x$

So, multiply the original ODE by x on both sides:

$(x^3+xy^2+x^2)\,\mathrm dx+x^2y\,\mathrm dy=0$

Now

$\dfrac{\partial(x^3+xy^2+x^2)}{\partial y}=2xy$

$\dfrac{\partial(x^2y)}{\partial x}=2xy$

so the modified ODE is exact.

Now we look for a solution of the form $F(x,y)=C$ , with differential

$\mathrm dF=\dfrac{\partial F}{\partial x}\,\mathrm dx+\dfrac{\partial F}{\partial y}\,\mathrm dy=0$

The solution F satisfies

$\dfrac{\partial F}{\partial x}=x^3+xy^2+x^2$

$\dfrac{\partial F}{\partial y}=x^2y$

Integrating both sides of the first equation with respect to x gives

$F(x,y)=\dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3+f(y)$

Differentiating both sides with respect to y gives

$\dfrac{\partial F}{\partial y}=x^2y+\dfrac{\mathrm df}{\mathrm dy}=x^2y$

$\implies\dfrac{\mathrm df}{\mathrm dy}=0\implies f(y)=C$

So the solution to the ODE is

$F(x,y)=C\iff \dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3+C=C$

$\implies\boxed{\dfrac{x^4}4+\dfrac{x^2y^2}2+\dfrac{x^3}3=C}$

5 0

3 years ago

Factorize: 3a^2 + 10a + 3

sergey [27]

Answer: (3a + 1) (a + 3)

Step-by-step explanation:

Concept:

Here, we need to know the idea of factorization.

It is like "splitting" an expression into a multiplication of simpler expressions. Factoring is also the opposite of Expanding.

Solve:

Given = 3a² + 10a + 3

STEP ONE: separate 3a² into two terms

STEP TWO: separate 3 into two terms

STEP THREE: match the four terms in ways that when doing cross-multiplication, the result will give us 10a.

3a 1

a 3

When cross multiply, 3a × 3 + 1 × a = 10a

STEP FOUR: combine the expression horizontally to get the final factorized expression.

3a ⇒ 1

a ⇒ 3

(3a + 1) (a + 3)

Hope this helps!! :)

Please let me know if you have any questions

7 0

3 years ago

Use the rules of inference to prove the conclusion r given (all 1,2,3 and 4) the four premises listed below. Write your solution

aniked [119]

Answer:

attached below

Step-by-step explanation:

Applying the rule of logical equivalences

attached below is a detailed solution ( written as a numbered sequence of statements )

3 0

2 years ago

If y = 12 x – 12, determine the value of y when x = –16.

GrogVix [38]

Answer:

Step-by-step explanation: y=180

4 0

3 years ago

Ms.Jackson went to the local market to buy 50 pieces of fruit. Out of the 50 pieces of fruit she bought, 14 were apples. What fr

harkovskaia [24]

Answer:

Good luck kid mark me most brainliest!

8 0

3 years ago

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