Exercise
$\frac{dy}{dx}-2y=\left(xy\right)^5$
Step-by-step Solution
Learn how to solve problems step by step online. Solve the differential equation dy/dx-2y=(xy)^5. The power of a product is equal to the product of it's factors raised to the same power. We identify that the differential equation \frac{dy}{dx}-2y=x^5y^5 is a Bernoulli differential equation since it's of the form \frac{dy}{dx}+P(x)y=Q(x)y^n, where n is any real number different from 0 and 1. To solve this equation, we can apply the following substitution. Let's define a new variable u and set it equal to. Plug in the value of n, which equals 5. Simplify.
Solve the differential equation dy/dx-2y=(xy)^5
Final answer to the exercise
$\frac{e^{8x}}{y^{4}}=-4\left(\frac{1}{8}x^5e^{8x}-\frac{5}{64}x^{4}e^{8x}+\frac{5}{128}x^{3}e^{8x}-\frac{15}{1024}x^{2}e^{8x}+\frac{15}{4096}xe^{8x}-\frac{15}{32768}e^{8x}\right)+C_0$