(2xy - 3x^{2})dx + (x^{2} + y)dy = 0

Donald Johnson

Donald Johnson

Answered question



Answer & Explanation

Stella Calderon

Stella Calderon

Beginner2021-12-22Added 35 answers

The given differential equation is
Comparing it with Mdx+Mdy=0
M=2xy3x2 N=x2+y
dMdy=2x dNdx=2x
Here, dMdy=dNdx
Then, it is exact differential equation, so solution is
Mdx+N (term not contining x) dy=c
This is solution of given DE.
Charles Benedict

Charles Benedict

Beginner2021-12-23Added 32 answers

Assuming (2xy3x2)dx+(x2+y)dy=0
Then 2xy+(x2+y)dydx=3x2 (1)
Notice that dx2y+y22dx is:
=2xy+(x2+y)dydx which is left hand side of (1)
So integrating (1) gives
So y2+2x2y2x3+c=0 is the solution


Expert2021-12-29Added 656 answers

Let M=(2xy3x2) so that DM/Dy=2x
Let N=:(x2+y) so that DN/Dx=2x
since DM/Dy=DN/Dx here D is to be read as delta.
The given Equation
(2xy3x2)dx+(x2+y)dy=0 is exact
Integrating yx2x6x+1/2y2=0
or, 2yx2x12x+y2=0

Nick Camelot

Nick Camelot

Skilled2023-05-23Added 164 answers

First, we check if the equation is exact by verifying if My=Nx, where M=2xy3x2 and N=x2+y.
Calculating the partial derivatives:
Since My=Nx, the equation is exact.
To find the solution, we integrate M with respect to x while treating y as a constant, and then find the potential function ϕ(x,y):
where f(y) is an arbitrary function of y.
Next, we differentiate ϕ(x,y) with respect to y and equate it to N:
Comparing the equation above with N=x2+y, we can see that dfdy=0.
Integrating dfdy=0 with respect to y, we obtain f(y)=C, where C is a constant.
Therefore, the potential function ϕ(x,y) is:
To find the final solution, we set ϕ(x,y) equal to a constant, which gives us:
Mr Solver

Mr Solver

Skilled2023-05-23Added 147 answers

Step 1: Identify the equation
The given differential equation is:
Step 2: Separate the variables
To solve the equation, we need to separate the variables x and y.
Step 3: Integrate both sides
Now, we integrate both sides of the equation.
Step 4: Evaluate the integrals
Integrating the left side with respect to x gives:
Using the power rule for integration, we have:
where C1 is the constant of integration.
Similarly, integrating the right side with respect to y gives:
where C2 is another constant of integration.
Step 5: Equate the integrals
Setting the integrals equal to each other, we have:
Simplifying the equation, we get:
Step 6: Combine the constants
Combining the constants C1 and C2 into a single constant C, we have:
Rearranging the equation, we obtain the solution:
Simplifying further, we get:
Therefore, the solution to the given differential equation is:
where C is the constant of integration.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?