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Exercise

$\frac{dh}{dw}\:h=4p^3+10p^2-3p-7,\:p=\frac{4}{3w-5}$

Step-by-step Solution

1

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

$h\cdot dh=\frac{-2880}{3w-5}dw$
2

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

$\int hdh=\int\frac{-2880}{3w-5}dw$
3

Solve the integral $\int hdh$ and replace the result in the differential equation

$\frac{1}{2}h^2=\int\frac{-2880}{3w-5}dw$
4

Solve the integral $\int\frac{-2880}{3w-5}dw$ and replace the result in the differential equation

$\frac{1}{2}h^2=-960\ln\left|3w-5\right|+C_0$
5

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

$h=\sqrt{-1920\ln\left(3w-5\right)+C_1},\:h=-\sqrt{-1920\ln\left(3w-5\right)+C_1}$

Final answer to the exercise

$h=\sqrt{-1920\ln\left(3w-5\right)+C_1},\:h=-\sqrt{-1920\ln\left(3w-5\right)+C_1}$

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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