Solve the differential equation

Evaluate

$$e=-\frac{dv}{dr}$$

Rewrite the expression

$$-\frac{dv}{dr}=e$$

Transform the expression

$$-dv=edr$$

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

$$\int -1dv=\int edr$$

Calculate

More Steps

Evaluate

$$\int -1dv$$

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

$$-v$$

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

$$-v+C_1,C_1\in \mathbb{R}$$

$$-v+C_1=\int edr,C_1\in \mathbb{R}$$

Calculate

More Steps

Evaluate

$$\int edr$$

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

$$er$$

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

$$er+C_2,C_2\in \mathbb{R}$$

$$-v+C_1=er+C_2,C_1\in \mathbb{R},C_2\in \mathbb{R}$$

$$\text{Since the integral constants }{C}_1\text{ and }{C}_2\text{ are arbitrary constants, replace them with constant C}$$

$$-v=er+C,C\in \mathbb{R}$$

Solution

$$v=-er+C,C\in \mathbb{R}$$

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