https://file.aiquickdraw.com/mathbot/file/1726435079763mz177qd5.jpg

To find a particular solution to the given differential equation 

\[
xy' = y + \left(\frac{2x^4}{y}\right) \cos(x^2),
\]

with the initial condition 

\[
y\left(\sqrt{\frac{\pi}{2}}\right) = \sqrt{\pi},
\]

we can approach this step-by-step. Here's how to solve it:

### Step 1: Simplify the Differential Equation
First, let's rewrite the differential equation in a more standard form:
\[
xy' - y = \frac{2x^4}{y} \cos(x^2).
\]

### Step 2: Introduce a Substitution
Given the structure of this equation, consider using a substitution to simplify. One potential substitution is \( y = x^2 f(x) \), as this might cancel some terms. Let's define:
\[
y = x^2 f(x).
\]

### Step 3: Differentiate the Substitution
Differentiate \( y = x^2 f(x) \) with respect to \( x \):
\[
y' = 2x f(x) + x^2 f'(x).
\]

### Step 4: Substitute into the Original Differential Equation
Substitute \( y \) and \( y' \) into the original differential equation:
\[
x[2x f(x) + x^2 f'(x)] = x^2 f(x) + \frac{2x^4}{x^2 f(x)} \cos(x^2).
\]

Simplify this expression:
\[
2x^2 f(x) + x^3 f'(x) = x^2 f(x) + \frac{2x^4 \cos(x^2)}{x^2 f(x)}.
\]

\[
2x^2 f(x) + x^3 f'(x) = x^2 f(x) + \frac{2x^2 \cos(x^2)}{f(x)}.
\]

### Step 5: Isolate Terms
Bring like terms together:
\[
x^3 f'(x) = -x^2 f(x) + \frac{2x^2 \cos(x^2)}{f(x)},
\]

which simplifies to:
\[
x f'(x) = -f(x) + \frac{2 \cos(x^2)}{f(x)}.
\]

### Step 6: Separation of Variables
Separate the variables if possible. This equation can be rearranged to:
\[
\frac{f'(x)}{-f(x) + \frac{2 \cos(x^2)}{f(x)}} = \frac{1}{x}.
\]

This is a complicated expression, so solving this exactly may require more advanced techniques or numerical methods.

### Step 7: Apply Initial Condition
Now, use the initial condition \( y\left(\sqrt{\frac{\pi}{2}}\right) = \sqrt{\pi} \) to find the particular solution:
\[
\sqrt{\frac{\pi}{2}}^2 f\left(\sqrt{\frac{\pi}{2}}\right) = \pi f\left(\sqrt{\frac{\pi}{2}}\right) = \sqrt{\pi}.
\]

Thus,
\[
f\left(\sqrt{\frac{\pi}{2}}\right) = \frac{1}{\sqrt{\pi}}.
\]

### Final Step: Particular Solution
Given the complexity of this differential equation, finding a closed-form solution may not be straightforward. It may require numerical methods to solve completely. However, this is the setup and how you would approach finding the particular solution.

Would you like further steps on numerical methods for this differential equation, or any clarification on the process so far?

### Further Questions
1. How would you approach solving this differential equation using numerical methods?
2. Can you explain the choice of substitution and its impact on simplifying the equation?
3. What other substitutions could potentially simplify this differential equation?
4. What role does the initial condition play in finding the particular solution?
5. How can we check if a function is a solution to a differential equation?

### Tip
When dealing with complex differential equations, it's often helpful to look for substitutions or transformations that simplify the terms. Numerical methods like Euler's or Runge-Kutta can be useful when analytic solutions are hard to find.

Find a particular solution of x*y' = y + (2*x^4/y) cos(x^2), y(sqrt(pi/2)) = sqrt(pi).

Explore the step-by-step solution to the non-linear differential equation xy' = y + (2x^4/y) cos(x^2) using substitution and separation of variables. This advanced calculus problem includes concepts from differential equations and trigonometry, suitable for university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation