Paritial derivative.

Homework Statement

Verify if ##t=\lambda x## then ##x^2\frac{\partial^2 y}{\partial x^2} = t^2\frac{\partial^2 y}{\partial t^2}##

The Attempt at a Solution

$$t=\lambda x\;\Rightarrow\; \frac{\partial t}{\partial x}=\lambda$$
$$\frac{\partial y}{\partial x} = \frac{\partial y}{\partial t}\frac{\partial t}{\partial x}= \lambda\;\frac{\partial y}{\partial t}$$
$$\frac{\partial^2 y}{\partial x^2}=\lambda \frac{\partial^2 y}{\partial t^2} \frac{\partial t}{\partial x}=\lambda^2\frac{\partial^2 y}{\partial t^2}$$

$$x^2\frac{\partial^2 y}{\partial x^2}=\frac{t^2}{\lambda^2}\lambda^2\frac{\partial^2 y}{\partial t^2}\;\Rightarrow\;x^2\frac{\partial^2 y}{\partial x^2} = t^2\frac{\partial^2 y}{\partial t^2}$$

Am I correct?

Thanks

Related Calculus and Beyond Homework Help News on Phys.org
arildno
Homework Helper
Gold Member
Dearly Missed
Yes, with just a minor quibble:
"t" is only dependent on "x", and not in addition dependent on other variables.
Thus, you should use dt/dx, rather than the symbol for the partial derivative here.

HallsofIvy
Homework Helper
Your differential equation has dependent variable "y" so it makes no sense to ask if "$t= \lambda x$" satisfies the equation.

I suspect you are asked to show that $f(t-\lambda x)$ satisfies the equation for f any twice differentiable function of a single variable.

arildno