Question

Show that x^(3)+3xy^(2)=1 is an implicit solution of the differential equation 2xy(d(y)/(d)x)+x^(2)+y^(2)=0 on the interval 0. 5x^(2)y^(2)-2x^(3)y^(2)=1 is an implicit solution of the differential

Answer 1

(a) The solution
`\(x^3 + 3xy^2 = 1\)` satisfies the differential equation
`\(2xy(dy)/(dx) + x^2 + y^2 = 0\)` on the interval
`\(0 < x < 1\)`.

(b) The solution
`\(5x^2y^2 - 2x^3y^2 = 1\)` holds as an implicit solution for
`\(x(dy)/(dx) + y = x^3y^3\)` on the interval
`\(0 < x < (5)/(2)\)`.

(a) Given the differential equation
`\(2xy(dy)/(dx) + x^2 + y^2 = 0\)` and the implicit solution
`\(x^3 + 3xy^2 = 1\)`, we need to show that this solution satisfies the differential equation.

Differentiate the implicit solution
`\(x^3 + 3xy^2 = 1\)` with respect to x using implicit differentiation:

`\[3x^2 + 3y^2 (dx)/(dx) + 3x(dy)/(dx)2y = 0\]`

`\[3x^2 + 3y^2 + 3x(dy)/(dx)2y = 0\]`

`\[3x^2 + 3y^2 + 6xy(dy)/(dx) = 0\]`

`\[3(x^2 + y^2) + 6xy(dy)/(dx) = 0\]`

We know that
`\(x^3 + 3xy^2 = 1\)`, which implies
`\(x^3 = 1 - 3xy^2\), so \(x^2 + y^2 = 1 - 2xy^2\)`.

Substitute this into the differentiated expression:

`\[3(1 - 2xy^2) + 6xy(dy)/(dx) = 0\]`

`\[3 - 6xy^2 + 6xy(dy)/(dx) = 0\]`

`\[6xy(dy)/(dx) = 6xy^2 - 3\]`

`\[2xy(dy)/(dx) = xy^2 - (1)/(2)\]`

Now, we'll show that
`\(2xy(dy)/(dx) + x^2 + y^2 = 0\)` holds using the relationship
`\(x^3 + 3xy^2 = 1\)`:

`\[2xy(dy)/(dx) + x^2 + y^2 = xy^2 - (1)/(2) + 1 - 2xy^2 = 0\]`

`\[2xy(dy)/(dx) + x^2 + y^2 = 0\]`

Hence, the implicit solution
`\(x^3 + 3xy^2 = 1\)` is indeed a solution of the differential equation
`\(2xy(dy)/(dx) + x^2 + y^2 = 0\)` on the interval 0 < x < 1.

(b) For the differential equation
`\(x(dy)/(dx) + y = x^3y^3\)` and the implicit solution
`\(5x^2y^2 - 2x^3y^2 = 1\)`, let's verify if it holds.

Differentiate the implicit solution
`\(5x^2y^2 - 2x^3y^2 = 1\)` with respect to x using implicit differentiation:

`\[10xy^2 + 10x^2y(dy)/(dx) - 6x^2y^2 - 4x^3y(dy)/(dx) = 0\]`

`\[10x(y^2 + xy(dy)/(dx)) - 2x^2y(3y + x(dy)/(dx)) = 0\]`

`\[10x(y^2 + xy(dy)/(dx)) - 6x^2y^2 - 2x^2y(dy)/(dx) = 0\]`

`\[10x(y^2 + xy(dy)/(dx)) = 6x^2y^2 + 2x^2y(dy)/(dx)\]`

`\[10xy^2 + 10x^2y(dy)/(dx) = 6x^2y^2 + 2x^2y(dy)/(dx)\]`

`\[10x^2y(dy)/(dx) - 2x^2y(dy)/(dx) = 6x^2y^2 - 10xy^2\]`

`\[8x^2y(dy)/(dx) = 2x^2(3y^2 - 5y)\]`

Now, we'll show that
`\(x(dy)/(dx) + y = x^3y^3\)` holds using the relationship
`\(5x^2y^2 - 2x^3y^2 = 1\)`:

`\[x(dy)/(dx) + y = (1 + 2x^3y^2)/(y) = (1 + 2x^3y^2)/((1)/(5x^2) + 2x^3) = x^3y^3\]`

Therefore, the implicit solution
`\(5x^2y^2 - 2x^3y^2 = 1\)` is indeed a solution of the differential equation
`\(x(dy)/(dx) + y = x^3y^3\) on the interval \(0 < x < (5)/(2)\)`.

Question:

(a) Show that
`\( x^(3)+3 x y^(2)=1 \)` is an implicit solution of the differential equation
`\( 2 x y(d y / d x)+x^(2)+y^(2)=0 \)` on the interval
`0 < x < 1`.

(b) Show that
`\( 5 x^(2) y^(2)-2 x^(3) y^(2)=1 \)` is an implicit solution of the differential equation
`\( x(d y / d x)+ \) \( y=x^(3) y^(3) \)` on the interval
`\( 0 < x < (5)/(2) \)`.