Bernoulli differential equations
E582378
first-order nonlinear differential equation
named mathematical concept
type of ordinary differential equation
Bernoulli differential equations are a class of first-order nonlinear differential equations that can be transformed into linear form and are fundamental in the study of ordinary differential equations.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Bernoulli differential equation | 0 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
first-order nonlinear differential equation
ⓘ
named mathematical concept ⓘ type of ordinary differential equation ⓘ |
| appearsIn |
courses on differential equations
ⓘ
introductory ODE textbooks ⓘ |
| becomesLinearFor |
n = 0
ⓘ
n = 1 ⓘ |
| canBeTransformedInto | linear first-order differential equation ⓘ |
| category | equations solvable by integrating factor ⓘ |
| classification | special first-order ODE ⓘ |
| conditionOnExponent |
n ≠ 0
ⓘ
n ≠ 1 ⓘ |
| domain |
complex-valued functions
ⓘ
real-valued functions ⓘ |
| field |
analysis
ⓘ
differential equations ⓘ mathematics ⓘ |
| generalSolutionForm | depends on integral of transformed linear equation ⓘ |
| hasOrder | first order ⓘ |
| hasParameter | exponent n ⓘ |
| historicalPeriod | late 17th century ⓘ |
| involvesFunction |
P(x)
ⓘ
Q(x) ⓘ |
| involvesUnknownFunction | y(x) ⓘ |
| involvesVariable | x ⓘ |
| isA | nonlinear ordinary differential equation ⓘ |
| isNonlinearFor | n ≠ 0,1 ⓘ |
| isTaughtAt | undergraduate level ⓘ |
| namedAfter | Jacob Bernoulli NERFINISHED ⓘ |
| prerequisiteKnowledge |
basic differential equations
ⓘ
calculus ⓘ |
| property |
nonlinear due to power of y
ⓘ
reducible to linear form by substitution ⓘ |
| relatedTo |
Riccati differential equations
NERFINISHED
ⓘ
linear first-order differential equations ⓘ |
| requiresCondition | P(x) and Q(x) continuous on interval of interest ⓘ |
| requiresStep |
apply integrating factor to transformed equation
ⓘ
divide by y^n (for y ≠ 0) ⓘ |
| solutionMethod |
integrating factor method
ⓘ
reduction to linear equation ⓘ |
| solutionSpace | one-parameter family of solutions ⓘ |
| solvedBy |
substitution u = y^{1-n}
ⓘ
substitution v = y^{1-n} ⓘ |
| standardForm | y' + P(x)y = Q(x)y^n ⓘ |
| usedIn |
chemical kinetics
ⓘ
fluid mechanics ⓘ growth and decay processes ⓘ modeling population dynamics ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.