Find y(x) and f(y) given y''(x)=f(y) and int(f(y)dy)=x-1

  • Thread starter Thread starter Bill_Nye_Fan
  • Start date Start date
Bill_Nye_Fan
Messages
31
Reaction score
2

Homework Statement


A function of y, ##f(y)##, is known to be equal to the second derivative of function ##y(x)##. ( i.e. ##\frac{d^2y}{dx^2}=f\left(y\right)## )

Given that ##\int _{ }^{ }f\left(y\right)dy=x-1##, and function ##y(x)## has a stationary point at ##x=1## and an x-intercept at ##x=2##, find ##y(x)## and hence find ##f(y)##


Homework Equations


Knowledge of solving differential equations.

The Attempt at a Solution


##\int _{ }^{ }f\left(y\right)dy=x-1##
##\frac{d^2y}{dx^2}=f\left(y\right)##
##∴ \int _{ }^{ }\frac{d^2y}{dx^2}dy=\int _{ }^{ }\frac{d\left(\frac{dy}{dx}\right)}{dx}dy=x-1##
I derived both sides with respect to ##y##
##\frac{d\left(\frac{dy}{dx}\right)}{dx}=\frac{d\left(x-1\right)}{dy}##
And this is where I get stuck. I'm not sure what to do from here, can someone please help? Thank you.
 
Physics news on Phys.org
In
$$
\int _{ }^{ }\frac{d\left(\frac{dy}{dx}\right)}{dx}dy = \int \frac{dy}{dx} d\left(\frac{dy}{dx}\right)
$$
if you write ##dy/dx = v##, can you do the integral analytically?
 
blue_leaf77 said:
In
$$
\int _{ }^{ }\frac{d\left(\frac{dy}{dx}\right)}{dx}dy = \int \frac{dy}{dx} d\left(\frac{dy}{dx}\right)
$$
if you write ##dy/dx = v##, can you do the integral analytically?

Ah yes I see what you're saying. Integrating a function (or in this case a derivative function) with respect to itself is the same as integrating a singular variable with respect to itself. I've done some quick working out and this is what I got:
20170823_131847.jpg

Do you think this looks right or am I way off track?
 
If you check the consistency of your final result with the properties it should have as given in the problem, I guess then your answer is right.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top