Linda Seales

Answered

2021-12-27

Solve the differential equations:

$(1+{x}^{2}+{y}^{2}+{x}^{2}{y}^{2})dy={y}^{2}dx$

Answer & Explanation

Daniel Cormack

Expert

2021-12-28Added 34 answers

Simplify the equation into the first order separable ODE

$(1+{x}^{2}+{y}^{2}+{x}^{2}{y}^{2})dy={y}^{2}dx$ (1)

$(1+{x}^{2})(1+{y}^{2})\frac{dy}{dx}={y}^{2}$

$\frac{(1+{y}^{2})}{{y}^{2}}\frac{dy}{dx}=\frac{1}{(1+{x}^{2})}$

$(\frac{1}{{y}^{2}}+1)\frac{dy}{dx}=\frac{1}{(1+{x}^{2})}$

$(\frac{1}{{y}^{2}}+1)dy=\frac{1}{{x}^{2}+1}dx$

Integrate of both side

$\int (\frac{1}{{y}^{2}}+1)dy=\int \frac{1}{{x}^{2}+1}dx$

$\int \frac{1}{{y}^{2}}dy+\int 1\cdot dy=\int \frac{1}{{x}^{2}+1}dx$

$\frac{-1}{y}+y={\mathrm{tan}}^{-1}\left(x\right)+C$ (2)

Solve for y

$\frac{-1}{y}+y={\mathrm{tan}}^{-1}\left(x\right)+C$

$\frac{{y}^{2}-1}{y}={\mathrm{tan}}^{-1}\left(x\right)+C$

${y}^{2}-1=({\mathrm{tan}}^{-1}\left(x\right)+C)y$

${y}^{2}-({\mathrm{tan}}^{-1}\left(x\right)+C)y-1=0$

We get the quadratic equation,

${y}^{2}-({\mathrm{tan}}^{-1}\left(x\right)+C)y-1=0$ (3)

The standard form of the solution of a quadratic equation$(a{x}^{2}+bx+c)$ is given as

Integrate of both side

Solve for y

We get the quadratic equation,

The standard form of the solution of a quadratic equation

poleglit3

Expert

2021-12-29Added 32 answers

Simplifying

$(1+{x}^{2}+{y}^{2}+{x}^{2}{y}^{2})\cdot dy={y}^{2}dx$

Reorder the terms:

$(1+{x}^{2}+{x}^{2}{y}^{2}+{y}^{2})\cdot dy={y}^{2}dx$

Reorder the terms for easier multiplication:

$dy(1+{x}^{2}+{x}^{2}{y}^{2}+{y}^{2})={y}^{2}dx$

$(1\cdot dy+{x}^{2}\cdot dy+{x}^{2}{y}^{2}\cdot dy+{y}^{2}\cdot dy)={y}^{2}dx$

Reorder the terms:

$({dx}^{2}y+{dx}^{2}{y}^{3}+1dy+{dy}^{3})={y}^{2}dx$

$({dx}^{2}y+{dx}^{2}{y}^{3}+1dy+{dy}^{3})={y}^{2}dx$

Solving

$dx}^{2}y+{dx}^{2}{y}^{3}+1dy+{dy}^{3}=dx{y}^{2$

Solving for variable d.

Move all terms containing d to the left, all other terms to the right.

Add$-1left.dxright.{y}^{2}$ to each side of the equation.

$dx}^{2}y+{dx}^{2}{y}^{3}+1dy+-1dx{y}^{2}+{dy}^{3}=dx{y}^{2}+-1dx{y}^{2$

Reorder the terms:

Reorder the terms:

Reorder the terms for easier multiplication:

Reorder the terms:

Solving

Solving for variable d.

Move all terms containing d to the left, all other terms to the right.

Add

Reorder the terms:

Vasquez

Expert

2022-01-09Added 457 answers

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