Solve the differential equation

Evaluate

$$df=fdr$$

Swap the sides

$$fdr=df$$

Rewrite the expression

$$dr=\frac{1}{f}\times df$$

$$\text{Integrate the left-hand side of the equation with respect to }r\text{ and the right-hand side of the equation with respect to }f$$

$$\int 1dr=\int \frac{1}{f}df$$

Calculate

More Steps

Evaluate

$$\int 1dr$$

$$\text{Use the property of integral }\int kdx=kx$$

$$r$$

$$\text{Add the constant of integral }{C}_1$$

$$r+C_1,C_1\in \mathbb{R}$$

$$r+C_1=\int \frac{1}{f}df,C_1\in \mathbb{R}$$

Calculate

More Steps

Evaluate

$$\int \frac{1}{f}df$$

$$\text{Use the property of integral }\int \frac{1}{x}dx=\ln \left|x\right|$$

$$\ln \left(f\right)$$

$$\text{Add the constant of integral }{C}_2$$

$$\ln \left(f\right)+C_2,C_2\in \mathbb{R}$$

$$r+C_1=\ln \left(f\right)+C_2,C_1\in \mathbb{R},C_2\in \mathbb{R}$$

Solution

$$r=\ln \left(f\right)+C,C\in \mathbb{R}$$

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