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Exercise

$\frac{dp}{dt}=\left(1-2t\right)\left(p-2\right)$

Step-by-step Solution

1

Group the terms of the differential equation. Move the terms of the $p$ variable to the left side, and the terms of the $t$ variable to the right side of the equality

$\frac{1}{p-2}dp=\left(1-2t\right)dt$
2

Integrate both sides of the differential equation, the left side with respect to $p$, and the right side with respect to $t$

$\int\frac{1}{p-2}dp=\int\left(1-2t\right)dt$
3

Expand the integral $\int\left(1-2t\right)dt$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int\frac{1}{p-2}dp=\int1dt+\int-2tdt$
4

Solve the integral $\int\frac{1}{p-2}dp$ and replace the result in the differential equation

$\ln\left|p-2\right|=\int1dt+\int-2tdt$
5

Solve the integral $\int1dt+\int-2tdt$ and replace the result in the differential equation

$\ln\left|p-2\right|=t-t^2+C_0$

Final answer to the exercise

$\ln\left|p-2\right|=t-t^2+C_0$

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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Main Topic: Limits to Infinity

The limit of a function f(x) when x tends to infinity is the value that the function takes as the value of x grows indefinitely.

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