Question

Solution of \(\left(d^{2} y / d x^{2}\right)=\log x\) is: (A) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (B) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}+(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (C) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}-\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (D) None of these

Short answer

The solution of the given differential equation is: \(y = \frac{1}{2}x^2 \log x - \frac{3}{4}x^2 + C_1x + C_2\).

Step-by-step solution

Integrate the differential equation once

Let's integrate the given differential equation with respect to x once: \[\int \frac{d^2 y}{dx^2} dx = \int \log x dx\] Let the first derivative of y, \(\frac{dy}{dx}\), be denoted by \(v\). Then we have: \[\frac{dy}{dx} = v = \int \log x dx + C_1\]

Integrate the differential equation again

Now we need to integrate the equation we derived above: \[\int \frac{dy}{dx} dx = \int v dx = \int \left( \int \log x dx + C_1 \right) dx\] Using integration by parts, we can find the value of \(\int \log x dx\): Let u = \(\log x\) and \(dv = dx\), then: \(du = \frac{1}{x} dx\) and \(v = x\). Now, applying integration by parts: \[\int \log x dx = x \log x - \int x (\frac{1}{x}dx)\] \[\int \log x dx = x \log x - \int dx\] \[\int \log x dx = x \log x - x + C\] Now, substitute this value back into the equation: \[y = \int (x \log x - x + C + C_1) dx\] \[y = \frac{1}{2}x^2 \log x - \frac{1}{2}x^2 + Cx + C_1x + C_2\] Comparing this with the given options:

Compare the derived equation with given options

The derived equation matches option (A): \(y = \frac{1}{2}x^2 \log x - \frac{3}{4}x^2 + C_1x + C_2\) So the correct solution is Option (A).

Key concepts

Integration by Parts

Integration by Parts is a useful mathematical technique. It helps in finding the integral of a product of two functions. The method is derived from the product rule of differentiation. It can simplify complex integrals that are hard to solve using basic rules. Here's an easy way to use Integration by Parts:

• Choose two functions, usually denoted as \( u \) and \( dv \). These functions are picked from the integral of the product \( \int u \, dv \).
• The rules for choosing \( u \) and \( dv \) are guided by the LIATE rule. LIATE stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. Choose \( u \) according to this order of priority.
• Differentiate \( u \) to find \( du \), and integrate \( dv \) to find \( v \).

The formula of Integration by Parts is given as: \[\int u \, dv = uv - \int v \, du\]In the original problem, this technique was used to integrate \( \log x \). Identifying \( u = \log x \) and \( dv = dx \), the method simplified the integral to facilitate solving the differential equation. This set the path for solving the second-order differential equation, illustrating the power and necessity of mastering Integration by Parts.

Second Order Differential Equations

Second Order Differential Equations involve derivatives up to the second order. They frequently emerge in fields like physics, engineering, and other sciences, describing the behavior of systems over time or distance. The general form is:\[\frac{d^2y}{dx^2} + P(x) \frac{dy}{dx} + Q(x)y = R(x)\]The given problem belongs to a simpler sub-category where only the second derivative equals a function, i.e., \( \frac{d^2y}{dx^2} = \log x \). Solving these equations involves two main steps:

• First Integration: This reduces the differential order from second to first. You integrate \( \log x \) considering constants of integration like \( C_1 \).
• Second Integration: This step takes the first derivative (after the first integration) to the original function \( y \). Here, another constant \( C_2 \) emerges.

Integration methods like Integration by Parts are indispensable here. Through solving such equations, we can determine the motion in dynamic systems and model real-world phenomena accurately.

Integration Techniques

Integration Techniques are a broad set of methods employed to solve integrals of various kinds. In the context of solving differential equations, these techniques are critical. Choosing the right technique can simplify the problem drastically. Here are some prominent methods employed in typical scenarios:

• Basic Integration: Direct application of antiderivatives, such as the integral of a polynomial.
• Integration by Parts: Useful for integrating the product of two functions as discussed earlier. It's crucial to solve forms like \( \int x \log x \) efficiently.
• Partial Fraction Decomposition: It breaks down complex rational expressions into simpler fractions that are easier to integrate.
• Substitution: Often called \( u \)-substitution, simplifies integrals by changing variables to achieve an easier form.

While dealing with the exercise, the focus was on using Integration by Parts to handle \( \log x \). However, recognizing and applying suitable integration techniques are essential skills in solving complex mathematical problems, including differential equations.