Question

solve the deferential equation dy/dx=4x^4y^4 with the condition that y(2)=2

the solution is y=________?

Answer 1

This is what you call a "separable" differential equation, means expressions involving y and x may be separated into factors.

Let's demonstrate... also, pretend that dy and dx are expressions in their own right and may be multiplied, divided, or cancelled...

\[\Large \frac{\color{red}{dy}}{\color{blue}{dx}} = 4x^4y^4\]
Multiply both sides by dx, the left-side dx cancels, giving you...
\[\Large \color{red}{dy} = 4x^4y^4\color{blue}{dx}\]
Next, divide both sides by y^4, cancelling out the y^4 on the left
\[\Large \frac1{y^4}\color{red}{dy}=4x^4\color{blue}{dx}\]

then just integrate both sides...\[\Large \int\frac1{y^4}\color{red}{dy}=\int4x^4\color{blue}{dx}\]

Answer 2

i saw what you did there... ;)

Answer 3

so -4/y^5=4x^5/5

Answer 4

?