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Bernoulli Differential Equations
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Standard form of a Bernoulli equation
The standard form is . To solve, use the substitution which turns it into a linear differential equation.
Basic Bernoulli equation
Use the substitution . The equation becomes , which has a solution . Hence, .
Bernoulli equation with exponential functions
Substitute . The transformed equation is , which can be solved using an integrating factor to find . Then reverse the substitution for .
Solve
Apply the substitution . The new equation can be solved. The final solution for involves combining constants and applying the initial conditions.
Handling an equation of the form
Use the substitution to linearize the equation to . Solve for using integrating factor or other means, then find from .
Bernoulli equation example with trigonometric functions
The substitution linearizes the equation to . After finding , the solution for is found by reversing the substitution.
Transforming to solve for
Substitute which converts the equation to . Solve the linear ODE for and then reverse the substitution to find .
Solving
Take as the substitution, which turns the equation into . Solve for using an integrating factor, then find by transforming back from .
Working with a negative exponent
Substitute to get a linear equation . Solve for , and then calculate given that .
General solution approach for
The substitution transforms the Bernoulli equation into a linear equation . After finding , solve for using the relationship between and .
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