anonymous
  • anonymous
solve the DE: dy/dx=-(y^3 +(4e^x)y)/2e^x + 3y^2)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
can anyone help? i am stuck at trying the separation of variables
anonymous
  • anonymous
want variables for d and e right?
anonymous
  • anonymous
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

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anonymous
  • anonymous
(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
anonymous
  • anonymous
i am completely lost with what you did
anonymous
  • anonymous
i am telling you how to get each variable
anonymous
  • anonymous
its differential equations so i am only solving for y in the end
anonymous
  • anonymous
okay so need y
anonymous
  • anonymous
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
anonymous
  • anonymous
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
anonymous
  • anonymous
maybe this will clear it up some

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