Question

$y^{\prime \prime }+4x^{2}y=0$

Short answer

Solve the characteristic equation for r and form the general solution y(x).

Step-by-step solution

Identify the type of differential equation

Determine that the given equation is a second-order linear differential equation. The equation is given by y'' + 4x^2y = 0.

Assume a solution form

Assume that the solution to the differential equation is of the form y = e^{r x^2}.This form is chosen due to the presence of the variable coefficient 4x^2.

Compute the first and second derivatives

Calculate the first derivative y' = d(e^{r x^2})/dx = 2r x e^{r x^2},and the second derivative y'' = d(2r x e^{r x^2})/dx = 2r e^{r x^2} + 4r^2 x^2 e^{r x^2}.

Substitute back into the differential equation

Substitute the original assumed solution and its derivatives back into the differential equation:y'' + 4x^2y = (2r e^{r x^2} + 4r^2 x^2 e^{r x^2}) + 4 x^2 e^{r x^2} = 0.

Simplify the equation

Combine like terms and factor out common terms:e^{r x^2}(2r + 4r^2 x^2 + 4x^2) = 0.Since e^{r x^2} ≠ 0, set the remaining factor to zero: 2r + 4r^2 x^2 + 4x^2 = 0.

Solve for r

Rearrange the equation to solve for r: 2r + 4x^2(r^2 + 1) = 0.Divide by 2:r + 2x^2(r^2 + 1) = 0.

Determine possible values of r

This step might include solving a quadratic equation (a characteristic equation) or any other relevant algebra to determine feasible values for r.

General solution

Based on the root values of r, construct the general solution y(x).The form will generally be different based on whether the roots are real and distinct, real and repeated, or complex conjugates.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. These equations are fundamental in describing various phenomena in engineering, physics, economics, and other sciences.

Here are some key points about differential equations:

• Differential equations can be classified based on order and linearity.
• The order of a differential equation is determined by the highest derivative present in the equation.
• A linear differential equation means the function and its derivatives appear to the power of one, and they aren’t multiplied together.

The equation we’re dealing with here is a second-order linear differential equation, as the highest derivative is the second derivative ( y'' ) and all terms are linear in y and its derivatives.

Second-Order Linear Differential Equations

Second-order linear differential equations involve the first and second derivative of a function. They generally take the form:

a(x)y'' + b(x)y' + c(x)y = f(x)

Key characteristics include:

• They describe systems with more complex behaviors than first-order equations.
• Depending on the values of a(x), b(x), and c(x), the methods to solve these equations can vary.

The given equation is y'' + 4x^2 y = 0 , which fits the standard form where a(x)=1 , b(x)=0 , c(x)=4x^2 , and f(x)=0 (homogeneous).

Solution Methods

To solve second-order linear differential equations, several methods exist. For homogeneous equations like our example, common methods include:

• Using characteristic equations
• Reduction of order
• Series solutions

In our problem, we hypothesize a solution of a specific form y = e^{rx^2} due to the variable coefficient 4x^2 present in the differential equation. This assumption helps simplify the equation so it can be solved more readily. Once you assume the solution form, take the derivatives, substitute them back into the equation, and solve for the constant parameters.

Characteristic Equations

In solving second-order linear differential equations, the characteristic equation plays a key role, especially for homogeneous cases. It’s derived from the assumed solution. In our exercise:

• We assumed y = e^{r x^2} .
• The derivatives are calculated and substituted back into the differential equation.

This substitution and reorganization lead us to:

e^{rx^2}(2r + 4r^2x^2 + 4x^2) = 0

Since e^{rx^2} is never zero, the term in the parentheses must equal zero, creating our characteristic equation:
2r + 4r^2 x^2 + 4x^2 = 0

Solving it helps us identify possible values of r, crucial for forming the general solution to the differential equation.