Solving a Bernoulli Differential Equation by Taking the Reciprocal

Summary

The document presents a Bernoulli differential equation with a quadratic term in the unknown function and asks how to solve it. It describes a substitution that takes the reciprocal of the function, differentiates that reciprocal, and converts the original nonlinear equation into a first-order linear equation. This is a useful general technique: an appropriate change of variables can make a nonlinear differential equation easier to solve using standard linear methods.

The post does not include the solution of the resulting equation or discuss initial conditions, so it gives no worked result to check. The reciprocal substitution also assumes the original function is nonzero on the interval where it is applied; any solution with zeros or special parameter cases may need separate consideration. The author additionally requests reading suggestions on stochastic control and advanced algebra, but the document provides no recommendations or application to a trading model.

Key ideas

• Taking the reciprocal transforms the stated nonlinear equation into a first-order linear equation.
• The derivative of the reciprocal follows from the chain rule.
• A linear first-order equation can be solved with standard integrating-factor methods.
• The reciprocal substitution requires care where the original function is zero.
• The document poses the solution and book recommendations as questions and does not supply them.

Tags

# Help in Bernoulli's differential equation


# Help in Bernoulli's differential equation












I want to solve the following Bernoulli differential equation: $$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$ where $\sigma$ and $a$ are real numbers.

Until now I have divided both sides of the equation with $A^{2}(t)$ and defined $u=A^{-1}(t)$ and $u'=-A^{-2}(t)A'(t)$. The new equation that arises is $$u'+2au=1-2\sigma $$

It must be relatively simple but how do I solve this? Can someone give me some hints? Also can someone propose some good books about stochastic control and advanced algebra (including Ricatti equations and differential equations etc)?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.