Solving a Separable Differential Equation

Question

The table outlines the steps for solving the differential equation $$\frac{dy}{dx} = \frac{y}{x}$$ with the initial condition $$y(1)=e$$. All of the following statements about this process are true EXCEPT:

Step | Operation
:---: | :---:
1 | Separate variables: $$\frac{dy}{y} = \frac{dx}{x}$$
2 | Integrate: $$ln|y| = ln|x| + C$$
3 | Exponentiate: $$y = Kx$$
4 | Apply $$y(1)=e$$ to get $$K = e$$

Accepted Answer

After integration, the general solution is $$y = ln|x| + C$$, which appropriately represents the antiderivative.

Answer

Exponentiating both sides yields the general solution $$y = K*x$$, and applying the initial condition gives $$K = e$$.

Answer

Separating variables yields $$\frac{dy}{y} = \frac{dx}{x}$$, an essential first step.

Answer

Integrating both sides leads to $$ln|y| = ln|x| + C$$.

Step	Operation
1	Separate variables: $\frac{dy}{y} = \frac{dx}{x}$
2	Integrate: $ln\|y\| = ln\|x\| + C$
3	Exponentiate: $y = Kx$
4	Apply $y(1)=e$ to get $K = e$

Solving a Separable Differential Equation

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