Question

solve the DE:
dy/dx=-(y^3 +(4e^x)y)/2e^x + 3y^2)

Answer 1

can anyone help? i am stuck at trying the separation of variables

Answer 2

want variables for d and e right?

Answer 3

well separation of variables would be to the the dx and all x's on one side and the dy and all y's on the other

Answer 4

(dy)/(dx)=-(y^(3)+(4e^(x))*y)/(2)*e^(x)+3y^(2)

Multiply each term in the equation by d.
(dy)/(dx)*d=(-(y^(3)+(4e^(x))*y)/(2)*e^(x)+3y^(2))*d

Simplify the left-hand side of the equation by canceling the common terms.
(dy)/(x)=(-(y^(3)+(4e^(x))*y)/(2)*e^(x)+3y^(2))*d

Simplify the right-hand side of the equation by simplifying each term.
(dy)/(x)=-(dy(y^(2)e^(x)+4e^(2x)-6y))/(2)

Multiply each term in the equation by x.
(dy)/(x)*x=-(dy(y^(2)e^(x)+4e^(2x)-6y))/(2)*x

Simplify the left-hand side of the equation by canceling the common terms.
dy=-(dy(y^(2)e^(x)+4e^(2x)-6y))/(2)*x

Simplify the right-hand side of the equation by simplifying each term.
dy=-(dxy(y^(2)e^(x)+4e^(2x)-6y))/(2)

Multiply each term in the equation by 2.
dy*2=-(dxy(y^(2)e^(x)+4e^(2x)-6y))/(2)*2

Multiply dy by 2 to get 2dy.
2dy=-(dxy(y^(2)e^(x)+4e^(2x)-6y))/(2)*2

Simplify the right-hand side of the equation by simplifying each term.
2dy=-dxy(y^(2)e^(x)+4e^(2x)-6y)

Multiply -dxy by each term inside the parentheses.
2dy=(-dxy^(3)e^(x)-4dxye^(2x)+6dxy^(2))

Remove the parentheses around the expression -dxy^(3)e^(x)-4dxye^(2x)+6dxy^(2).
2dy=-dxy^(3)e^(x)-4dxye^(2x)+6dxy^(2)

Since d is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.
-dxy^(3)e^(x)-4dxye^(2x)+6dxy^(2)=2dy

Since 2dy contains the variable to solve for, move it to the left-hand side of the equation by subtracting 2dy from both sides.
-dxy^(3)e^(x)-4dxye^(2x)+6dxy^(2)-2dy=0

Factor out the GCF of -dy from each term in the polynomial.
-dy(xy^(2)e^(x))-dy(4xe^(2x))-dy(-6xy)-dy(2)=0

Factor out the GCF of -dy from -dxy^(3)e^(x)-4dxye^(2x)+6dxy^(2)-2dy.
-dy(xy^(2)e^(x)+4xe^(2x)-6xy+2)=0

If any individual factor on the left-hand side of the equation is equal to 0, the entire expression will be equal to 0.
-dy=0_(xy^(2)e^(x)+4xe^(2x)-6xy+2)=0

Set the first factor equal to 0 and solve.
-dy=0

Divide each term in the equation by -1y.
-(dy)/(-1y)=(0)/(-1y)

Simplify the left-hand side of the equation by canceling the common terms.
d=(0)/(-1y)

Any expression with zero in the numerator is zero.
d=0

Set the next factor equal to 0 and solve.
(xy^(2)e^(x)+4xe^(2x)-6xy+2)=0

Remove the parentheses around the expression xy^(2)e^(x)+4xe^(2x)-6xy+2.
xy^(2)e^(x)+4xe^(2x)-6xy+2=0

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
ln(xy^(2)e^(x)+4xe^(2x)-6xy+2)=ln(0)

The logarithm of 0 is undefined.
Undefined

Since the logarithm is undefined, there is no solution.
No Solution

The final solution is all the values that make -dy(xy^(2)e^(x)+4xe^(2x)-6xy+2)=0 true.
d=0

Answer 5

i am completely lost with what you did

Answer 6

i am telling you how to get each
variable

Answer 7

its differential equations so i am only solving for y in the end

Answer 8

okay so need y

Answer 9

dy/dx= 2-e^x/ 3+2y

this can be done using variable separable method; collect x and dx terms along one side and y and dy terms along other side
hence
(3+2y)dy= (2-e^x)dx
integrating both sides
3y +2y^2/2 = 2x - e^x +c
3y+y^2= 2x-e^x+c
so now y^2+3y -2x+e^x=c ------A is the solution where c is the integrative constant

in next question do you want to find max value for dy/dx or y
if you want to find max value for y then equate dy/dx =0
so 2-e^x/3+2y=0
2-e^x=0
e^x=2 ==> x= log 2
now the intervals ae -infinity to log2 and log2 to infinity
now substitute values in between -ve inf to log2 in A and check if it is less than 0 then that value is max else it is max
sim do for the values between log2 to infinty

Answer 10

dy/dx = (2 - e^x)/(3 + 2y)

(3 + 2y) dy = (2 - e^x) dx

Integrating both sides:

3y + y² = 2x - e^x + C

(y + 3/2)² = 2x - e^x + C

y = +/-√(2x - e^x + C) - 3/2

Answer 11

maybe this will clear it up some