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Exercise

$\frac{u^2-1}{u}dy=2ydu$

Step-by-step Solution

1

Divide fractions $\frac{1}{\frac{u^2-1}{u}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$\frac{u}{u^2-1}=\frac{1}{2y}$
2

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

$\int\frac{u}{u^2-1}du=\int\frac{1}{2y}dy$
3

Solve the integral $\int\frac{u}{u^2-1}du$ and replace the result in the differential equation

$\frac{1}{2}\ln\left|u+1\right|+\frac{1}{2}\ln\left|u-1\right|=\int\frac{1}{2y}dy$
4

Solve the integral $\int\frac{1}{2y}dy$ and replace the result in the differential equation

$\frac{1}{2}\ln\left|u+1\right|+\frac{1}{2}\ln\left|u-1\right|=\frac{1}{2}\ln\left|y\right|+C_0$
5

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

$u=\sqrt{C_2y+1},\:u=-\sqrt{C_2y+1}$

Final answer to the exercise

$u=\sqrt{C_2y+1},\:u=-\sqrt{C_2y+1}$

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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