Solve the differential equation

Evaluate

$$y^{\prime }=y$$

Rewrite the expression

$$\frac{dy}{dx}=y$$

Rewrite the expression

$$\frac{1}{y}\times \frac{dy}{dx}=y\times \frac{1}{y}$$

Multiply the terms

$$\frac{1}{y}\times \frac{dy}{dx}=1$$

Transform the expression

$$\frac{1}{y}\times dy=dx$$

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

$$\int \frac{1}{y}dy=\int 1dx$$

Calculate

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Evaluate

$$\int \frac{1}{y}dy$$

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

$$\ln \left(y\right)$$

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

$$\ln \left(y\right)+C_1,C_1\in \mathbb{R}$$

$$\ln \left(y\right)+C_1=\int 1dx,C_1\in \mathbb{R}$$

Calculate

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Evaluate

$$\int 1dx$$

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

$$x$$

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

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

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

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

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

$$\text{Convert the logarithm into exponential form using the fact that }{\log }_{a}x=b\text{ is equal to }x=a^b$$

$$y=e^{x+C},C\in \mathbb{R}$$

Solution

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Evaluate

$$e^{x+C}$$

$$\text{Use }{a}^{m+n}=a^m\times a^n\text{ to expand the expression}$$

$$e^C\times e^x$$

$$\text{Since the expression }{e}^C\text{ is a constant,it is possible to denote that whole expression as a constant C}$$

$$C{e}^x$$

$$y=C{e}^x,C\in \mathbb{R}$$

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