Question

Finding general solutions Find the general solution of each differential equation. Use \(C, C_{1}, C_{2}, \ldots\) to denote arbitrary constants. $$y^{\prime \prime}(x)=\frac{x}{\left(1-x^{2}\right)^{3 / 2}}$$

Short answer

The general solution to the given differential equation is: $$y(x) = \arcsin(x) + C_1x + C_2$$

Step-by-step solution

Integrate the given equation once

Since we're given the second derivative, \(y^{\prime \prime}(x)\), let's integrate once with respect to \(x\) to find the first derivative, \(y'(x)\): $$y'(x) = \int y^{\prime \prime}(x) dx = \int \frac{x}{\left(1-x^{2}\right)^{3 / 2}} dx$$ This integral is non-trivial, so we will use a substitution method. Let \(u = 1-x^2\), then \(du = -2x dx\). We have: $$y'(x) = \int \frac{-1}{2} \cdot \frac{-2x dx}{u^{3/2}} = - \frac{1}{2} \int u^{-\frac{3}{2}} du$$ Now we can integrate \(u\): $$y'(x) = - \frac{1}{2}\left[- \frac{2}{\sqrt{u}}\right] + C_1 = \frac{1}{\sqrt{1-x^2}} + C_1$$ Here, we have integrated once and obtained \(y'(x)\).

Integrate again to find y(x)

We now need to find \(y(x)\) by integrating the expression for \(y'(x)\): $$y(x) = \int y'(x) dx = \int \left(\frac{1}{\sqrt{1-x^2}} + C_1\right) dx$$ We will split this integral into two parts and integrate each separately: $$y(x) = \int \frac{1}{\sqrt{1-x^2}} dx + C_1 \int dx$$ First, we can integrate the second part easily, and it results in \(C_1x\). Now we need to solve the first integral, which is a standard integral. Recall that \(\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin(x)\). Thus, we have: $$y(x) = \arcsin(x) + C_1x + C_2$$ Here, we have found the general solution to the given differential equation: $$y(x) = \arcsin(x) + C_1x + C_2$$

Key concepts

Integration Techniques

Understanding integration techniques is crucial when solving differential equations. The process of finding the antiderivative, or integral, is often not straightforward and can involve several methods. Sometimes we can use direct integration if the function is a standard form for which we know the antiderivative. However, more complex functions require creative techniques like substitution, integration by parts, partial fraction decomposition, or even numerical methods when an analytical solution is difficult to obtain.

When tackling the integral of a complex function, it's imperative to identify which technique will simplify the problem. For instance, when encountering an integrand with a square root in the denominator, like \( \frac{x}{(1-x^{2})^{\frac{3}{2}}} \), the substitution method often proves useful. In our case, choosing a suitable substitution can help us transform the integrand into a more manageable form and reveal a straightforward path to the solution.

Second Derivative

The second derivative of a function, denoted as \( y''(x) \) or \( \frac{d^{2}y}{dx^2} \), tells us a lot about the function’s curvature. It essentially describes how the rate of change of the function's slope is itself changing. A thorough understanding of second derivatives is vital when solving second-order differential equations.

In the presented exercise, the given equation is a second-order differential equation, indicating that it represents the second derivative of an unknown function \( y(x) \). To solve for \( y(x) \) from its second derivative, we perform integration twice. This process moves us ‘backwards’ from acceleration to velocity (first derivative), and then from velocity to position (the function \( y(x) \)).

Arcsin Function

The arcsin function, also known as the inverse sine function, is a fundamental function in trigonometry. It is the inverse operation of the sine function, designated as \( \arcsin(x) \) or \( \sin^{-1}(x) \). The arcsin function determines the angle whose sine is the given value, which implies it takes an input between -1 and 1, and returns an angle in radians (or degrees) within the interval \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

In relation to integrals involving square roots and binomials such as \( \frac{1}{\sqrt{1-x^2}} \), knowing that the antiderivative is \( \arcsin(x) \) simplifies the solution process. As showcased in our problem, this knowledge helps us to directly write down part of the general solution for the differential equation.

Substitution Method

The substitution method, often employed in integrating complex functions, involves replacing a part of the integrand with a new variable. This can transform a difficult integral into an easier one. This technique requires two main steps: first, identify an appropriate substitution that simplifies the integral, and second, perform the substitution correctly to find the new limits of integration if necessary.

In our original exercise, we use the substitution \( u = 1 - x^2 \) for the integral of the second derivative. By differentiating this expression with respect to \( x \), we obtain \( du = -2x dx \) which allows us to replace \( x dx \) in the original integrand. This transforms the complicated integral into a simpler form \( u^{-\frac{3}{2}} \), which we can integrate easily. The substitution method is a powerful tool for making seemingly intractable integrals approachable.