Solve the following differential equation ;

Question

y'' = (4/x)y + x(y^1/2)

Answer 1

To solve the differential equation

\[ y'' = \frac{4}{x} y + x y^{1/2}, \]

we can start by rearranging the equation for clarity:

\[ y'' - \frac{4}{x} y - x y^{1/2} = 0. \]

This is a second-order nonlinear ordinary differential equation (ODE). The presence of the term \(y^{1/2}\) suggests that a substitution may be useful.

Let’s try a substitution. Set

\[ y = v^2, \]

where \(v\) is a function of \(x\). Then, we have:

\[ y' = 2v v' \quad \text{and} \quad y'' = 2(v'v' + vv''). \]

Substituting \(y\) in the differential equation gives:

\[ 2(v'v' + vv'') = \frac{4}{x}v^2 + x v. \]

Now we can rearrange this into:

\[ 2(v'v' + vv'') - \frac{4}{x}v^2 - xv = 0. \]

This differential equation may still be complex to solve directly, so let’s explore different methods by checking particular solutions.

Let's consider a power solution of the form \(y = kx^n\). Substitute \(y = kx^n\) into the original equation and find \(k\) and \(n\). We will compute derivatives to see if we can balance the equation based on the degrees of \(x\).

Calculating \(y'\) and \(y''\):

\[ y' = knx^{n-1}, \quad y'' = kn(n-1)x^{n-2}. \]

Plugging into the equation gives:

\[ kn(n-1)x^{n-2} = \frac{4}{x} (kx^n) + x(kx^{n/2}). \]

This simplifies to:

\[ kn(n-1)x^{n-2} = 4k x^{n-1} + kx^{n/2 + 1}. \]

Equating powers of \(x\) on both sides can lead to equations for \(k\) and \(n\). You may also start testing integer values.

1. Suppose \(n = 2\):
   • This gives \(y = kx^2\).
   • The terms then reduce similar to powers leading to

\[ 2k = 4k \quad \text{which implies a contradiction unless k=0.} \]

2. Try \(n = 1\):
   • This gives \(y = kx^1\),
   • This leads to \(2k = 4k + k(1) = 5k\).
   • A contradiction here as well unless \(k = 0\).

Given the complexity, numerical or qualitative solutions might be more efficient based on boundary conditions or specific initial values.

In some cases, transformations or numerical methods like Runge-Kutta or shooting methods might be employed if you'd like to explore specific solution branches.

Please provide any additional conditions you want me to include or if something more specific is desired for resolving this DE!

Answer 2

To solve the given differential equation

y'' = \frac{4}{x}y + x y^{1/2},
we can use a substitution or series solution method since it's not directly solvable by standard methods like separation of variables or integrating factor. Here, we'll attempt a solution using a series expansion:

Assume a series solution:
Let's assume
y(x)
can be expressed as a power series:
y(x) = \sum_{n=0}^{\infty} a_n x^n
Find derivatives:
First derivative:
y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1}
Second derivative:
y''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}
Substitute into the differential equation:
Substitute
y
:
y = \sum_{n=0}^{\infty} a_n x^n
Substitute
y'
:
y' = \sum_{n=1}^{\infty} n a_n x^{n-1}
Substitute
y''
:
y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}
The term
\frac{4}{x}y
becomes:
\frac{4}{x} \sum_{n=0}^{\infty} a_n x^n = 4 \sum_{n=0}^{\infty} a_n x^{n-1}
The term
x y^{1/2}
is more complex because
y^{1/2}
isn't straightforward in series form; however, for simplicity, we can approximate or handle it term by term if we assume
y \geq 0
:

Because of the term
y^{1/2}
, this approach becomes less straightforward, suggesting we need a different method or simplification.
Alternative Method:
Consider the equation in another form or use numerical methods since an analytical solution might not be simple.

Here's a conceptual approach:
If
y^{1/2}
can be approximated or if we focus on specific conditions (like near
x=0
or for small
y
), we might use:
y^{1/2} \approx a_0^{1/2} + \frac{1}{2} a_1 x + \text{higher order terms}
However, this equation might not have a straightforward closed-form solution due to the nonlinear term
x y^{1/2}
.

For a precise solution, you would typically:

Use numerical methods like Runge-Kutta for integrating this differential equation numerically.
Or, transform the equation if possible through some substitution or coordinate change to simplify.

Given the complexity and nonlinearity, the exact analytical solution might not be feasible without further context or simplification. If you need a more precise or numerical approach, please specify further conditions or constraints for
y
or
x
.