Question

Solve the differential equation. $$ \left(1+x^{2}\right) y^{2}-2 x y=0 $$

Short answer

Our final solution to the homogeneous differential equation is \(y=K/x\), where K is the constant of integration.

Step-by-step solution

Identify Equation Type

The given equation is \((1+x^{2})y^{2}-2 x y = 0\), which can be rewritten in the form of a homogeneous differential equation \(y'=(y/x)\). The equation becomes \(y'+(2/x)y=0\).

Implement Substitution

Let's use the substitution \(v=y/x\) or \(y=vx\). So, \(y'=v+xv'\). Substituting these into the equation gives: \(v+xv'+(2/x)vx=0\).

Separate Variables

After cleaning up, the equation becomes: \(xv'+2v=0\). Now, the variables can be separated by dividing both sides by \(xv\), resulting in \((v'/v) + 2/x = 0\).

Integrate Both Sides

The variables have been separated and the differential equation can be integrated. That is: \(\int dv/v = - \int 2/x dx\). After integration, we get \(\ln |v| = -2 \ln |x| + C\), where C is the constant of integration.

Exponentiate Both Sides

Now, exponentiate both sides to eliminate the natural logarithm, resulting in: \(|v| = |x|^{-2} e^C\). Therefore \(v=x^{-2}K\), where \(K=e^C\) which is a constant.

Substitute Back

Finally, recall the substitution from Step 2 that \(v=y/x\), now we can replace \(v\) back to the equation and find the solution for y. So, \(y/x = x^{-2}K\) or \(y= K/x\), this is the general solution to this problem.