True and False regarding differential equations.

Question

The question states If f(x) > 0 for all x, then every solution of the differential equation dy/dx = f(x)is an increasing function.

For this one I said False because even if f(x) is positive doesn't mean its increasing, but I got it wrong. I was just wondering if you could explain then why this would be true?

 

the second question says If the function y = f(x) is a solution of the differential equation dy/dx= (sin(x)/x), then the function y = f(x) + 5 is also a solution.

This one I don't really know, but if I had to guess I would say true because f(x) would just shift it by 5  but it would still be the answer right. Please help me, I am a little confused. Thank you in advance.

Answer 1

(1)

 

This is stating that the derivative is positive, which means the slope of the tangent line is positive, so the function must be increasing along the +x direction.  TRUE.

 

(2)

 

Work this one out explicitly.  If y = f(x) is a solution, then df(x)/dx = sin(x) / x.

 

d/dx [f(x) + 5] = df(x)/dx + 0 = sin(x) / x

 

Then f(x) + 5 is also a solution.  TRUE.  (You will get a family of solutions for any differential equation.  Some value f(x₀) at x₀ must be specified so you can use it to get a unique solution.)