y^{\frac{1}{2}}\,\frac{\mathrm{d}y}{\mathrm{d}x}+y^{\frac{3}{2}}=1,\;\;{x}_0=0,\;y\left(0\right)=4

To solve the differential equation \( y^{\frac{1}{2}} \frac{\mathrm{d}y}{\mathrm{d}x} + y^{\frac{3}{2}} = 1 \) with the initial condition \( y(0) = 4 \), we can proceed as follows:

### Step 1: Simplify the Equation
First, let's rewrite the equation in a more standard form:
\[
y^{\frac{1}{2}} \frac{\mathrm{d}y}{\mathrm{d}x} = 1 - y^{\frac{3}{2}}
\]
Now, divide both sides by \( y^{\frac{1}{2}} \) to separate variables:
\[
\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1 - y^{\frac{3}{2}}}{y^{\frac{1}{2}}}
\]

### Step 2: Separate Variables
To separate the variables, we rewrite the equation as:
\[
\frac{\mathrm{d}y}{1 - y^{\frac{3}{2}}} = \frac{\mathrm{d}x}{y^{\frac{1}{2}}}
\]

### Step 3: Integrate Both Sides
Integrate both sides with respect to their respective variables:

\[
\int \frac{\mathrm{d}y}{1 - y^{\frac{3}{2}}} = \int \frac{\mathrm{d}x}{y^{\frac{1}{2}}}
\]

However, this form is quite complex and typically requires special functions to solve analytically. Instead, let’s approach it by a substitution method.

### Step 4: Substitution to Simplify Integration
Let \( u = y^{\frac{1}{2}} \), then \( y = u^2 \) and \( \frac{\mathrm{d}y}{\mathrm{d}x} = 2u \frac{\mathrm{d}u}{\mathrm{d}x} \).

Substituting into the original equation:
\[
u \cdot 2u \frac{\mathrm{d}u}{\mathrm{d}x} + u^3 = 1
\]
\[
2u^2 \frac{\mathrm{d}u}{\mathrm{d}x} + u^3 = 1
\]
\[
2u^2 \frac{\mathrm{d}u}{\mathrm{d}x} = 1 - u^3
\]
\[
\frac{\mathrm{d}u}{\mathrm{d}x} = \frac{1 - u^3}{2u^2}
\]

### Step 5: Solve the Differential Equation
Now, separate variables and integrate:
\[
\int \frac{2u^2 \, \mathrm{d}u}{1 - u^3} = \int \mathrm{d}x
\]

This integral is still complex, but it can be approached using partial fraction decomposition or special functions. For now, let's express the solution implicitly, as:

\[
x + C = \int \frac{2u^2 \, \mathrm{d}u}{1 - u^3}
\]

### Step 6: Apply the Initial Condition
Using the initial condition \( y(0) = 4 \), we have \( u(0) = 2 \). So, substitute \( u = 2 \) and \( x = 0 \) into the implicit solution to find \( C \).

The integral and resulting expression are complex and might require numerical methods to fully solve. However, you can approximate or express the solution implicitly as done above.

Would you like a detailed numerical solution or further clarification on a specific part?

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Here are five questions related to this problem:
1. How can you solve the integral \(\int \frac{2u^2 \, \mathrm{d}u}{1 - u^3}\) using partial fractions?
2. What numerical methods can be applied to solve this differential equation?
3. How does the substitution \( u = y^{\frac{1}{2}} \) simplify the equation?
4. What are the challenges in solving nonlinear differential equations?
5. Can you find an approximate solution to this problem using a series expansion?

**Tip:** When dealing with complex integrals, consider substitution or numerical approaches to simplify the problem.

Learn how to solve the differential equation y^(1/2) dy/dx + y^(3/2) = 1 with an initial condition y(0) = 4. This problem involves using the separation of variables method and substitution techniques to find the solution. Suitable for university-level students and those studying differential equations.

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