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Exercise

$\frac{dv}{dt}=v^{\frac{3}{2}}$

Step-by-step Solution

1

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

$\frac{1}{\sqrt{v^{3}}}dv=dt$
2

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

$\int\frac{1}{\sqrt{v^{3}}}dv=\int1dt$
3

Solve the integral $\int\frac{1}{\sqrt{v^{3}}}dv$ and replace the result in the differential equation

$\frac{-2}{\sqrt{v}}=\int1dt$
4

Solve the integral $\int1dt$ and replace the result in the differential equation

$\frac{-2}{\sqrt{v}}=t+C_0$
5

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

$v=\frac{4}{\left(t+C_0\right)^{2}}$

Final answer to the exercise

$v=\frac{4}{\left(t+C_0\right)^{2}}$

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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