1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Arisa [49]
3 years ago
8

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

Mathematics
1 answer:
aksik [14]3 years ago
5 0

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

<em>is</em> 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 <em>x</em>, and dividing this by N(x,y)=xy gives an expression independent of <em>y</em>. If we assume \mu=\mu(x) is a function of <em>x</em> 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 <em>x</em> 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 <em>F</em> 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 <em>x</em> 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 <em>y</em> 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}

You might be interested in
34:45:63 = x:90:126<br> What is x?
e-lub [12.9K]

Answer:

x = 68

Step-by-step explanation:

If you divided 90 by 45, you can see that the 1st ratio has been multiplied by 2 to get the second ratio. To get x, simply multiply 34 by 2 and you should get 68

4 0
3 years ago
Read 2 more answers
The area of a parallelogram is 72 inches. what is the height?
bulgar [2K]
Area of a parallelogram is base x height
9x8=72
the height should be 8?
4 0
2 years ago
(38x+61y)-(3x-11y) simplify this expression
Roman55 [17]

━━━━━━━☆☆━━━━━━━

▹ Answer

<em>35x + 72y</em>

<em />

▹ Step-by-Step Explanation

(38x + 61y) - (3x - 11y)

38x + 61y - (3x - 11y)

38x + 61 y - 3x + 11y

35x + 61y + 11y

35x + 72y

Hope this helps!

- CloutAnswers ❁

Brainliest is greatly appreciated!

━━━━━━━☆☆━━━━━━━

4 0
3 years ago
What is the solution to the division problem below you can use long division or synthetic division 2x^3-2x^2-10x-6/ x-3
CaHeK987 [17]

Answer:

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Solve the system of linear equations below 2x-y=0 3x-2y=-3
Lynna [10]
2x - y = 0
2x = y

3x - 2(2x) = -3
3x - 4x = -3
-x = -3
x = 3

2(3) = y
6 = y
3 0
3 years ago
Other questions:
  • Eunice and Belinda weighs 267 pound 6 ounces. Eunice and Laura weighs 223 pounds and 8 ounces. Laura weighs 136 pounds and 11 ou
    13·1 answer
  • What is next number and why
    11·1 answer
  • A right rectangular prism has a length of 6 cm, a width of 8 cm, and a height of 4 cm. The dimensions of the prism are halved.
    11·1 answer
  • Find the value of x.
    11·1 answer
  • Which exponent is missing from this equation<br> 5.6=56,000
    6·1 answer
  • Determine the slope of the line passing through the given points.<br><br> (-2, -2) and (-2, 3)
    5·1 answer
  • We want to find the zeros of this polynomial:
    14·1 answer
  • Use the drawing tools to form the correct answer on the number line. Graph the solution set to this inequality
    8·2 answers
  • Find the greatest common factor of 7 and 7
    10·2 answers
  • Factor 81y+72.<br><br>Write your answer as a product with a whole number greater than 1.<br>help plz
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!