Math Problem Statement

Find the general solution for the differential equation dx/dt + (3/t)x + x^3t^2 = 0, t > 0.

Solution

To solve the given differential equation:

dxdt+3tx+x3t2=0,t>0,\frac{dx}{dt} + \frac{3}{t}x + x^3 t^2 = 0, \quad t > 0,

we proceed as follows:


Step 1: Identify the type of differential equation

This is a first-order non-linear differential equation due to the x3x^3 term. However, it can be solved using substitution because it appears separable in terms of tt and xx.


Step 2: Rewrite the equation

dxdt=3txx3t2.\frac{dx}{dt} = -\frac{3}{t}x - x^3 t^2.

This can be rewritten as: dxdt=x(3t+x2t2).\frac{dx}{dt} = -x \left(\frac{3}{t} + x^2 t^2 \right).


Step 3: Use substitution to simplify

Separate variables xx and tt: 1x(3t+x2t2)dx=dt.\frac{1}{x \left( \frac{3}{t} + x^2 t^2 \right)} dx = -dt.


Step 4: Simplify further

Perform integration on both sides. To handle the left-hand side integration, you can break it into partial fractions or substitution for x2x^2. We will solve this carefully to find the general solution.

Let me compute the integral. sympy import symbols, Function, Eq, dsolve, Derivative

Define the variables and function

t = symbols('t', positive=True) # t > 0 is given x = Function('x')(t) # x is a function of t

Define the differential equation

diff_eq = Eq(Derivative(x, t) + (3/t)x + x**3t**2, 0)

Solve the differential equation

general_solution = dsolve(diff_eq, x)

general_solutionThe general solution to the given differential equation is:

x(t)=±3t3(C12t3),x(t) = \pm \sqrt{\frac{3}{t^{3}(C_1 - 2t^3)}},

where C1C_1 is an arbitrary constant of integration.


Do you need a step-by-step explanation for solving this equation, or have additional questions? Here are some related questions:

  1. How does the substitution method simplify separable differential equations?
  2. Can we interpret this solution in terms of initial conditions?
  3. How do non-linear terms in xx affect the solution's behavior?
  4. What are the practical applications of such differential equations?
  5. How does t>0t > 0 influence the solution's domain?

Tip: Always verify the solution by substituting it back into the original equation to check if it satisfies the equation.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Separable Equations
Integration Techniques

Formulas

dx/dt + P(t)x = Q(t)x^n
Separable variables: 1/f(x) dx = g(t) dt

Theorems

Method for Solving First-Order Nonlinear Differential Equations
Integration of Rational Functions

Suitable Grade Level

Undergraduate Mathematics (First-Year Calculus)