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
5(x – 4) + 3x – 9x = 6 - (2x + 5) + 8x
AURORKA [14]
This is what i got:
-x-20=6x+1
6 0
3 years ago
The answer is 915 / 7
Naily [24]

Answer:

If you want 915/7 simplified, it is 130.714285714.

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Daphne wants to buy a coat that costs $80. The store has a sales tax of 6.5%. How much tax will Daphne be charged if she buys th
enot [183]

Answer:

$85.2

Step-by-step explanation:

6 0
3 years ago
In a town, 15% of the population is unemployed. what is the
kow [346]

Answer:

1479 people

Step-by-step explanation:

If the amount employed is 8381, that means 8381 is 85% of the population. 8381/85 = 98.6

Every 1% of the population is 98.6, so we can just do 15 x 98.6 to get 1479

7 0
2 years ago
Read 2 more answers
Last month maria high the 5 mile Mountain Trail a number of times and she hiked the 10 mile canal trail several times. Let X rep
Marrrta [24]

Alright, lets get started.

Maria hiked the 5 mile Mountain Trail a number of times and she hiked the 10 mile canal trail several times.

x times she hiked 5 mile mountain.

So, number of mils she hiked for 5 miles will be = 5 x

Similarly,

y times she hiked 10 mile canal,

So, number of miles she hiked for canal = 10 y

So, total number of miles she hiked = 5 x +10 y

But as per question given,  she had a total of 90 miles

It means

5x + 10 y = 90

Dividing this equation with 5

x + 2 y = 18

So, the equation is x + 2 y = 18 :   Answer

Hope it will help :)




8 0
3 years ago
Other questions:
  • Which statement is true ?
    12·1 answer
  • What is 6.04 in simplest form
    5·1 answer
  • The size of the largest angle in a triangle is 4 times the size of the smallest angle.
    8·1 answer
  • What is the simplified form of the following expression? sqrt 1/144
    10·1 answer
  • At a garage sale, Darnell buys 5 dozen golf balls for $45.00 and 2 dozen cans of tennis balls for $30.00. What is the unit price
    10·1 answer
  • Does this image have a vertical reflection, horizontal reflection, and rotation? I'll give brainiest!
    10·1 answer
  • Buying a car is an example of _____.
    7·1 answer
  • Eliminate the parameter in the equations x =t Superscript one-half and y = t – 4. How can the rectangular equation be described?
    11·2 answers
  • Select all the values of m that make the inequality m&lt;7.4 true. 7.40, 7.62, 6.8, 7.09, 8.1.
    5·2 answers
  • There are 454 grams in a line. There are 16 ounces in a pound.How many grams are in an ounce?
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!