Question

Find the general solution of the following differential equations. $$y^{\prime \prime}(x)=\frac{x}{\left(1-x^{2}\right)^{3 / 2}}$$

Short answer

Answer: The general solution of the given differential equation is \(y(x) = \frac{1}{2}(\arcsin{x} + \frac{1}{2}x\sqrt{1-x^2}) + C\), where \(C\) is an integration constant.

Step-by-step solution

Integrate the given equation with respect to x

To find the first derivative, y'(x), integrate both sides of the given equation with respect to x: $$\int y^{\prime\prime}(x) dx = \int \frac{x}{(1-x^2)^{3/2}} dx$$

Integrate RHS by substitution

To integrate the right-hand side (RHS) of the equation, we can make a substitution. Let \(u = 1-x^2\), so \( du = -2x dx\). Hence, we have $$\int y^{\prime\prime}(x) dx = -\frac{1}{2}\int \frac{1}{u^{3/2}} du$$

Solve the integral

Now, we can solve the integral on the RHS: $$-\frac{1}{2}\int \frac{1}{u^{3/2}} du = -\frac{1}{2}\int u^{-3/2} du$$ Calculate the integral: $$= -\frac{1}{2} \cdot \frac{u^{-1/2}}{-1/2} + C_1 = u^{1/2} + C_1$$ Replace \(u\) with the original expression in terms of x: $$= (1-x^2)^{1/2} + C_1$$

Integrate both sides again

Now, we will integrate both sides of the equation to find the general solution, y(x): $$\int y^\prime(x) dx = \int (1-x^2)^{1/2} + C_1 dx$$

Solve the integrals

For this step, we need to integrate the function in two parts: $$y(x) = \int (1-x^2)^{1/2} dx + \int C_1 dx$$ The first integral can be solved using trigonometric substitution and the second integral is simply \(C_1x\). Let's solve the first integral: Let \(x = \sin{\theta}\), then \( dx = \cos{\theta} d\theta\). Hence, we have $$\int (1-\sin^2{\theta})^{1/2} \cos{\theta} d\theta$$ As we have \(1-\sin^2{\theta} = \cos^2{\theta}\), the integral becomes $$\int \cos^2{\theta} d\theta$$ To solve this integral, we use the double angle formula: $$\cos^2{\theta} = \frac{1 + \cos{2\theta}}{2}$$ So, the integral becomes $$\int \frac{1+\cos{2\theta}}{2} d\theta = \frac{1}{2}\int (1+\cos{2\theta}) d\theta = \frac{1}{2}(\theta + \frac{1}{2}\sin{2\theta}) + C_2$$ Convert back to x: $$\frac{1}{2}(\arcsin{x} + \frac{1}{2}x\sqrt{1-x^2}) + C_2$$

Combine the results

Now, we can combine the results from Step 5: $$y(x) = \frac{1}{2}(\arcsin{x} + \frac{1}{2}x\sqrt{1-x^2}) + C_2 + C_1x$$

Write the general solution

Finally, we can rewrite the equation by combining constants \(C_2\) and \(C_1x\) into a new constant \(C\), which yields the general solution: $$y(x) = \frac{1}{2}(\arcsin{x} + \frac{1}{2}x\sqrt{1-x^2}) + C$$

Key concepts

Integration Techniques

Understanding integration techniques is key to solving differential equations. Integration involves finding a function whose derivative is the given function. In the context of the exercise, we started with a second-order derivative, so we needed to integrate twice.

Here’s a simple breakdown:

• First, integrate to reduce the order of the differential equation. We moved from a second derivative to a first derivative, and then to the original function.
• Each integration can simplify into a more familiar form using substitutions or other known methods, like basic integrals tables.

This technique involves systematic approaches to find antiderivatives, easing the process of solving complex equations.

Trigonometric Substitution

Trigonometric substitution is a special technique that simplifies certain integrals by substituting trigonometric functions for variables. In this exercise, it was used to solve an integral involving a square root.

To break it down:

• We set the variable equal to a trigonometric function, like setting \(x = \sin \theta\), because it makes use of identities, such as \(1 - \sin^2 \theta = \cos^2 \theta\).
• The integral becomes more straightforward to evaluate with these substitutions, as trigonometric identities simplify expressions.

This method is particularly useful for integrals involving roots and is a powerful approach to handle otherwise complex problems.

General Solution

The general solution of a differential equation represents a family of functions that includes all possible solutions related to the initial equation. It’s typically expressed with arbitrary constants that can be specified by initial or boundary conditions.

In this problem:

• We combined the results of several integration steps to reach a solution that included constants \(C_1\) and \(C_2\).
• The final expression, \(y(x) = \frac{1}{2}(\arcsin{x} + \frac{1}{2}x\sqrt{1-x^2}) + C\), shows the general form.

The general solution is crucial because it contains all possible specific solutions, offering a comprehensive answer before specific conditions are applied.

Constants of Integration

When solving differential equations, integrating introduces arbitrary constants, known as constants of integration. These constants give solutions flexibility, representing a family of curves rather than a single solution.

Here’s what to know:

• Each integration step typically introduces a new constant because differentiation looks for all possible functions having the same derivative.
• In this exercise, we ended up simplifying \(C_1\) and \(C_2\) into a single constant \(C\) as part of the general solution, reflecting how constants can often be combined for simplicity.

Understanding these constants is important, as they allow solutions to be tailored to specific scenarios, such as initial values or boundary conditions.