Question

\(y^{\prime 3}-y y^{\prime 2}-x^{2} y^{\prime}+x^{2} y=0\)

Short answer

Question: Determine the general solution of the given first-order, nonlinear ordinary differential equation \(y^{\prime 3}-y y^{\prime 2}-x^{2} y^{\prime}+x^{2} y=0\). Answer: The general solution for the given differential equation is \(-\frac{1}{2}(yy^{\prime})^{2} + y\ln\left|yy^{\prime} - y\right| - \ln\left|yy^{\prime} - y\right| + C_1 = \frac{1}{3}x^3 + C\), where \(C_1=C\). Keep in mind that we discarded the singular solution \(\tilde y=y\), which needs to be considered separately.

Step-by-step solution

Rewrite the equation in a convenient form

First, rescale the dependent variable \(\tilde y=yy^{\prime}\): \[\left(y^{\prime}\right)^3 - y\left(y^{\prime}\right)^2 - x^2 y^{\prime} + x^2 y= 0\Longrightarrow \left(\tilde y\right)^3 - \tilde y^2 - x^2\tilde y + x^2 y = 0.\] We obtain the following equation \[\tilde y^{3} - \tilde y^{2} = x^{2}(y-\tilde y).\] Now, let's separate the variables:

Separate variables

We can rewrite our equation as: \[\frac{\tilde y^{3} - \tilde y^{2}}{y-\tilde y} = x^{2}.\] Now we can separate the variables: \[\frac{\tilde y^{2}(\tilde y - 1)}{\tilde y - y} d\tilde y = x^{2} dx.\]

Integrate both sides

Now, we integrate both sides of the equation: \[ \int \frac{\tilde y^{2}(\tilde y - 1)}{\tilde y - y} d\tilde y = \int x^{2} dx .\] Notice that \(\tilde y=y\) is a singular solution of the above integral, meaning it cannot be integrated using traditional methods. We discard this solution for now and continue with the integration. \[-\int \frac{\tilde y^{2}(\tilde y -1 )}{y - \tilde y}d\tilde y = \frac{1}{3} x^3 + C.\]

Solve the integral

The integral on the left-hand side can be solved using partial fraction decomposition, giving us: \[-\int \frac{\tilde y^{2}(\tilde y-1)}{y-\tilde y} d\tilde y = -\int \left(\tilde y - \frac{y}{\tilde y - y} + \frac{1}{\tilde y - y}\right) d\tilde y.\] So, \[-\frac{1}{2}\tilde y^{2} + y\ln\left|\tilde y - y\right| - \ln\left|\tilde y - y\right| + C_1 = \frac{1}{3}x^3 + C.\] Finally, replacing \(\tilde y = yy^{\prime}\), we obtain: \[-\frac{1}{2}(yy^{\prime})^{2} + y\ln\left|yy^{\prime} - y\right| - \ln\left|yy^{\prime} - y\right| + C_1 = \frac{1}{3}x^3 + C.\] This equation represents the general solution for the given differential equation, where \(C_1=C\). Keep in mind that we discarded the singular solution \(\tilde y =y\), which needs to be considered separately.