Problem 12
Question
In Problems, use the indicated change of variable to find the general solution of the given differential equation on the interval \((0, \infty)\). $$ x^{2} y^{\prime \prime}+\left(\alpha^{2} x^{2}-\nu^{2}+\frac{1}{4}\right) y=0 ; \quad y=\sqrt{x} u(x) $$
Step-by-Step Solution
Verified Answer
Perform the change of variable and simplify the equation.
1Step 1: Substitute the Change of Variable
The given differential equation is \(x^{2} y^{\prime \prime}+\left(\alpha^{2} x^{2}-u^{2}+\frac{1}{4}\right) y=0\). We are given \(y=\sqrt{x} u(x)\). Substitute this into the differential equation. First, calculate \(y^{\prime}\) and \(y^{\prime \prime}\).Calculate \(y^{\prime} = \left(\sqrt{x}\right)^{\prime} u(x) + \sqrt{x} u^{\prime}(x)\), which simplifies to \(\frac{1}{2\sqrt{x}} u(x) + \sqrt{x} u^{\prime}(x)\).Next, calculate \(y^{\prime \prime}\):\(y^{\prime \prime} = \left(\frac{1}{2\sqrt{x}} u(x)\right)^{\prime} + \left(\sqrt{x} u^{\prime}(x)\right)^{\prime} = -\frac{1}{4x^{3/2}} u(x) + \frac{1}{\sqrt{x}} u^{\prime}(x) + \frac{1}{2\sqrt{x}} u^{\prime}(x) + \sqrt{x} u^{\prime \prime}(x)\).So, \(y^{\prime \prime}(x) = -\frac{1}{4x^{3/2}} u(x) + \frac{3}{2\sqrt{x}} u^{\prime}(x) + \sqrt{x} u^{\prime \prime}(x)\).
Key Concepts
Change of VariablesGeneral SolutionSecond-Order Differential Equation
Change of Variables
A change of variables can simplify complex differential equations, making them easier to solve. In this exercise, the original equation is transformed using the substitution \( y = \sqrt{x} u(x) \). This substitution requires finding the first and second derivatives of \( y \) with respect to \( x \).
Calculating these derivatives can be a little tricky but is manageable by breaking it down:
This method allows the transformation of the differential equation into a different form. Often, this new form is more familiar or manageable.
Calculating these derivatives can be a little tricky but is manageable by breaking it down:
- The first derivative \( y' \) involves two parts: the derivative of \( \sqrt{x} \) with respect to \( x \), and the derivative involving \( u(x) \).
- The second derivative \( y'' \) builds on \( y' \) and includes additional terms from both \( u'(x) \) and \( u''(x) \). This makes the equation more complex, so careful calculation is vital.
This method allows the transformation of the differential equation into a different form. Often, this new form is more familiar or manageable.
General Solution
Once a differential equation is simplified using the change of variable, the goal is to find its general solution. This solution includes all possible solutions that satisfy the differential equation.
In this case, after substituting to find \( u(x) \) in terms of \( x \), you'll be looking to solve the resulting equation. The general solution captures all potential behaviors of the system described by the original equation.
Achieving the general solution helps in understanding the behavior of the differential equation on the interval \((0, \infty)\).
In this case, after substituting to find \( u(x) \) in terms of \( x \), you'll be looking to solve the resulting equation. The general solution captures all potential behaviors of the system described by the original equation.
- The general solution gives a family of solutions, often involving arbitrary constants.
- These constants can be determined if additional conditions, known as boundary or initial conditions, are provided.
- Finding the general solution involves integrating the transformed equation, which means solving for \( u(x) \) and translating back to \( y(x) \) using the original substitution.
Achieving the general solution helps in understanding the behavior of the differential equation on the interval \((0, \infty)\).
Second-Order Differential Equation
This problem involves a second-order differential equation, characterized by the presence of a second derivative term \( y'' \). Second-order equations are common in mathematical modeling of physical systems, like oscillations or dynamics.
These equations can often be intimidating due to their complexity, yet they are fundamental in describing systems where the rate of change of a rate of change is important.
Understanding second-order differential equations opens the doors to solving more complex real-life problems. Learning their solutions gives insights into advanced topics in calculus and applied mathematics.
These equations can often be intimidating due to their complexity, yet they are fundamental in describing systems where the rate of change of a rate of change is important.
- Second-order differential equations can typically involve notions of acceleration or curvature.
- Solving such equations often requires specific techniques like factorization, characteristic equations, or changes of variables.
- The solution of the second-order equation is vital in many scientific fields, predicting behaviors over time or space.
Understanding second-order differential equations opens the doors to solving more complex real-life problems. Learning their solutions gives insights into advanced topics in calculus and applied mathematics.
Other exercises in this chapter
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Rewrite the given expression as a single power series whose general term involves \(x^{k}\). $$ \sum_{\substack{n=1 \\ \infty}}^{\infty} 2 n c_{n} x^{n-1}+\sum_
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Put the given differential equation into the form (3) for each regular singular point of the equation. Identify the functions \(p(x)\) and \(q(x)\). $$ x y^{\pr
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Use an appropriate infinite series method about \(x=0\) to find two solutions of the given differential equation. $$ y^{\prime \prime}-x^{2} y^{\prime}+x y=0 $$
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$$ x^{2} y^{\prime \prime}+\left(\alpha^{2} x^{2}-\nu^{2}+\frac{1}{4}\right) y=0 ; \quad y=\sqrt{x} u(x) $$
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