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Exercise

$\left(q+1\right)\left(\frac{dc}{dq}\right)=cq$

Step-by-step Solution

1

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

$\frac{1}{c}dc=\frac{q}{q+1}dq$
2

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

$\int\frac{1}{c}dc=\int\frac{q}{q+1}dq$
3

Solve the integral $\int\frac{1}{c}dc$ and replace the result in the differential equation

$\ln\left|c\right|=\int\frac{q}{q+1}dq$
4

Solve the integral $\int\frac{q}{q+1}dq$ and replace the result in the differential equation

$\ln\left|c\right|=q-\ln\left|q+1\right|+C_1$
5

Find the explicit solution to the differential equation. We need to isolate the variable $c$

$c=\frac{C_1e^q}{q+1}$

Final answer to the exercise

$c=\frac{C_1e^q}{q+1}$

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: Separable Differential Equations

A separable differential equation is one that can be solved by separating variables, that is, when all expressions involving a variable can be placed on one side of the equation, and expressions involving the other variable on the other side of the equation.

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