Question

Solve: dy/dx+ y = xy^3 .??

Answer 1

`y = pm 1/sqrt( x + 1/2 + C e^(2x))`

Explanation:

`dy/dx+ y = xy^3`

This is non-linear but we can make it linear using a Bernoulli substitution. Here we let:

• `z = 1/y^2 qquad qquad z' = - 2/y^3 y'`

Multiply both sides of the DE by `- 2/y^3`:

`underbrace(dy/dx * (- 2/y^3))_(= z')+ underbrace(y* (- 2/y^3))_(= - 2z) = underbrace(xy^3 * (- 2/y^3))_(= - 2x)`

For this, the integrating factor is: `exp(int (-2) dx) = e^(-2x)`

`e^(- 2x) (z' - 2z) [=(e^(- 2x) z)^'] = - 2xe^(- 2x)`

Integrating both sides, using IBP on RHS:

`e^(- 2x) z = int \ x \ d( e^(- 2x))`

`e^(- 2x) z = \ x \ e^(- 2x) - int dx \ e^(- 2x)`

`e^(- 2x) z = x e^(- 2x) + 1/2 \ e^(- 2x) + C`

`z = x + 1/2 + C e^(2x) qquad = 1/y^2`

`y = pm 1/sqrt( x + 1/2 + C e^(2x))`