Question

Solve the differential equation. \(\frac{d y}{d x}=\frac{x^{3}-21 x}{5+4 x-x^{2}}\)

Short answer

The solution will be of the form \( y = \int \frac{x^{3}-21 x}{5+4 x-x^{2}} dx \). To get the particular solution, one needs to evaluate this integral.

Step-by-step solution

Transform the given equation into the standard form

Rearrange the differential equation into the standard form of first order linear differential equations \( \frac{dy}{dx} = P(x)y + Q(x) \) where P(x) and Q(x) are functions of x only. The given equation is already in this form with \( P(x) = 0 \) and \( Q(x) = \frac{x^{3}-21 x}{5+4 x-x^{2}} \).

Compute the integrating factor

The next step is to calculate the integrating factor, which is \( e^{\int P(x) dx} \). Since \( P(x) = 0 \), it implies that the integrating factor is \( e^{\int 0 dx} = e^{0} = 1 \) .

Multiply both sides by the integrating factor

Multiplying both sides of the equation by the integrating factor (which is 1 in this case), the equation remains the same, \( \frac{dy}{dx} = \frac{x^{3}-21 x}{5+4 x-x^{2}} \).

Integrate both sides of the equation

Now, integrate both sides of the equation with respect to x, \( \int \frac{dy}{dx} dx = \int \frac{x^{3}-21 x}{5+4 x-x^{2}} dx \), which simplifies to \( y = \int \frac{x^{3}-21 x}{5+4 x-x^{2}} dx \) .

Solve the integral to find the solution

Finally, solving the integral on the right-hand side will give the solution to the differential equation. This part mainly depends on your calculus skills.