Solve the differential equations: (1+x2+y2+x2y2)dy=y2dx

Linda Seales

Linda Seales

Answered

2021-12-27

Solve the differential equations:
(1+x2+y2+x2y2)dy=y2dx

Answer & Explanation

Daniel Cormack

Daniel Cormack

Expert

2021-12-28Added 34 answers

Simplify the equation into the first order separable ODE
(1+x2+y2+x2y2)dy=y2dx (1)
(1+x2)(1+y2)dydx=y2
(1+y2)y2dydx=1(1+x2)
(1y2+1)dydx=1(1+x2)
(1y2+1)dy=1x2+1dx
Integrate of both side
(1y2+1)dy=1x2+1dx
1y2dy+1dy=1x2+1dx
1y+y=tan1(x)+C (2)
Solve for y
1y+y=tan1(x)+C
y21y=tan1(x)+C
y21=(tan1(x)+C)y
y2(tan1(x)+C)y1=0
We get the quadratic equation,
y2(tan1(x)+C)y1=0 (3)
The standard form of the solution of a quadratic equation (ax2+bx+c) is given as
poleglit3

poleglit3

Expert

2021-12-29Added 32 answers

Simplifying
(1+x2+y2+x2y2)dy=y2dx
Reorder the terms:
(1+x2+x2y2+y2)dy=y2dx
Reorder the terms for easier multiplication:
dy(1+x2+x2y2+y2)=y2dx
(1dy+x2dy+x2y2dy+y2dy)=y2dx
Reorder the terms:
(dx2y+dx2y3+1dy+dy3)=y2dx
(dx2y+dx2y3+1dy+dy3)=y2dx
Solving
dx2y+dx2y3+1dy+dy3=dxy2
Solving for variable d.
Move all terms containing d to the left, all other terms to the right.
Add 1left.dxright.y2 to each side of the equation.
dx2y+dx2y3+1dy+1dxy2+dy3=dxy2+1dxy2
Reorder the terms:
Vasquez

Vasquez

Expert

2022-01-09Added 457 answers

(1+x2+y2+x2y2)dy=y2dx(1+x2)(1+y2)dy=y2dx(1+y2)/y2dy=1/(1+x2)dx(1+y2)/y2dy=1/(1+x2)dx1/y2+1dy=1/(1+x2)dx1/y+y=arctan(x)+C

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