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Verify that, for $t>0, y(t)=\ln t$ is a solution to the differential equation$$2\left(\frac{d y}{d t}\right)^{3}=\frac{d^{3} y}{d t^{3}}$$

$y(t)=(n(t)$ is a solution

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

Differential Equations

Campbell University

Harvey Mudd College

University of Nottingham

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in this problem, we have 1/3 order differential equation for which we'd like to determine if the following function would be a solution, nor to work this out. Let's first take three derivatives on the right hand side for our scratch work. So we're given that the function is the natural log, a rhythm of tea, the first or derivative we can the right D y t t or y prime, if you prefer is one over tea. Next we have to go up to the third order derivative. So let's take the derivative of the previous function, which is due to Y. D. T. Two. We would get for this second or derivative negative one over t squared. Then finally, one more derivative D three. Why d t cubed would be equal to positive to divide by t to the power of three. So now we're ready to come back to the original third or differential equation and determine if we have a solution. First, let's look at the left hand side. The left hand side will be twice the first or derivative to the power of three. So be involving this derivative here, I will say for a solution. Since twice de y DT cubed is equal to twice by substitution one over tea to the power of three, our result becomes to over t cubed. Let's compare this to the right hand side will say. And since D three way d t cubed is equal to by our work over here two over t cubed This shows that are given function. Why of tea equals and natural log of tea is a solution.

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