Question

Determine the general solution of the given differential equation that is valid in any interval not including the singular point. \(2 x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0\)

Short answer

Answer: The general solution for y(x) is \(y(x)=Ax^5+Bx^{-3}+Cx^5\int x^{-6}ydx\), where A, B, and C are constants of integration.

Step-by-step solution

Transform the differential equation using substitution

Let's transform the given differential equation by substituting \(u = y'\). First, differentiate our substitution: \(u = y' \Rightarrow u' = y''\) Now substitute \(u\) and \(u'\) into the original differential equation: \(2x^2(u') - 4x(u) + 6y = 0\)

Solve the first-order equation for \(u\)

Now, let's rewrite the equation as a first-order linear equation in u: \(2x^2u' - 4xu = -6y\) The equation is not homogeneous, but we can solve it without knowing \(y\), so we will proceed to find \(u\). To do this, divide both sides by \(2x^2\): \(u' - \frac{4}{x}u = \frac{-6y}{2x^2}\) \(u' -\frac{4}{x}u = -\frac{3y}{x^2}\) Now, we'll use an integrating factor to solve the first-order linear equation. The integrating factor is given by \(e^{\int -\frac{4}{x} dx}\): \(e^{-4\ln |x|} = e^{-4\ln x}=x^{-4}\) Now multiply the differential equation by the integrating factor: \(x^{-4}(u' -\frac{4}{x}u) = x^{-4}(-\frac{3y}{x^2})\) This will simplify to \((x^{-4}u)'=-3x^{-6}y\)

Integrate both sides

Integrate both sides with respect to \(x\) to find a relationship between \(u\) and \(y\): \(\int(x^{-4}u)' dx = \int-3x^{-6}y dx\) \( x^{-4}u = -3\int x^{-6}y dx + C\) \(u = -3x^4\int x^{-6}y dx + Cx^4\) We know that \(u = y'\) from our initial substitution, so we have: \(y'=-3x^4\int x^{-6}y dx + Cx^4\)

Integrate again to find the general solution

Now we will integrate both sides once more to get the general solution for \(y\): \(y(x)=\int (-3x^4\int x^{-6}y dx + Cx^4) dx=Ax^5+Bx^{-3}+Cx^5\int x^{-6}ydx\) Where \(A\) and \(B\) are constants of integration. Now, we have found the general solution of the given differential equation that is valid in any interval not including the singular point.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are mathematical equations involving functions and their derivatives. They are called 'ordinary' to distinguish them from 'partial' differential equations, which involve partial derivatives of functions of multiple variables. An ODE involves derivatives with respect to only one independent variable.

Understanding ODEs is crucial in modeling the rates of change in various scientific and engineering fields. For example, they describe the motion of planets, the growth of populations, the cooling of hot objects, and the behavior of electrical circuits. The order of a differential equation corresponds to the highest derivative term in the equation. Finding the general solution of an ODE involves determining a function that satisfies the given equation across a certain domain, except where there may be singular points where the solution is undefined.

Integrating Factor

The integrating factor is a powerful technique used to solve first-order linear ordinary differential equations. The method involves multiplying the entire differential equation by a strategically chosen function, which simplifies the equation, often turning the left-hand side of the equation into the derivative of a product of two functions.

This chosen function is the integrating factor and typically depends solely on the independent variable. It's determined by an exponentiated integral of the coefficient of the first-order derivative in the original equation. Using the integrating factor can turn a seemingly complex problem into one that can be approached with basic integration, enabling us to solve for the unknown function in the differential equation.

Substitution Method

The substitution method is a technique frequently employed to simplify differential equations by reducing the order or by transforming the equation into a form that can be more easily managed.

By substituting one variable or expression for another, we can often convert a higher-order differential equation into a first-order equation or make a non-linear equation linear. The key to effective substitution is choosing a new variable that either reveals an inherent pattern within the differential equation or makes the equation separable. After solving for the substitute variable, we then substitute back to find the solution in terms of the original variables.

Linear Differential Equations

Linear differential equations are a class of differential equations where the dependent variable and its derivatives appear to the first power and are not multiplied together. Such equations can often be solved using direct integration, the integrating factor method, or by using the method of undetermined coefficients.

For first-order linear differential equations, the standard form is \(\frac{dy}{dx} + P(x)y = Q(x)\), where \(P(x)\) and \(Q(x)\) are functions of \(x\). The solution to these equations can often give insight into physical processes, such as decay rates and population dynamics. The ability to solve linear differential equations efficiently is an essential tool in the mathematical sciences.

Second-order Differential Equations

Second-order differential equations contain second derivatives and are prevalent in the world of physics and engineering to describe systems with accelerations or curvature dependent behaviors. Many physical laws and principles such as Newton's second law of motion and the equation for a simple harmonic oscillator are expressed as second-order differential equations.

Solving second-order differential equations can be more challenging than solving first-order ones. The general solution typically includes two arbitrary constants, corresponding to the two initial conditions needed to specify a unique solution for an initial value problem. Depending on the coefficients and whether the equation is homogeneous or nonhomogeneous, various methods like the characteristic equation approach, method of undetermined coefficients, or variation of parameters are used for solving these equations.