Solve the differential equation

Evaluate

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

Rewrite the expression

$$\frac{dy}{db}=yb$$

Rewrite the expression

$$\frac{dy}{db}=by$$

Rewrite the expression

$$\frac{1}{y}\times \frac{dy}{db}=by\times \frac{1}{y}$$

Multiply the terms

$$\frac{1}{y}\times \frac{dy}{db}=b$$

Transform the expression

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

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

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

Calculate

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Evaluate

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

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

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

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

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

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

Calculate

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Evaluate

$$\int bdb$$

$$\text{Use the property of integral }\int x^{n}dx=\frac{x^{n+1}}{n+1}$$

$$\frac{b^{1+1}}{1+1}$$

Add the numbers

$$\frac{b^2}{1+1}$$

Add the numbers

$$\frac{b^2}{2}$$

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

$$\frac{b^2}{2}+C_2,C_2\in \mathbb{R}$$

$$\ln \left(y\right)+C_1=\frac{b^2}{2}+C_2,C_1\in \mathbb{R},C_2\in \mathbb{R}$$

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

$$\ln \left(y\right)=\frac{b^2}{2}+C,C\in \mathbb{R}$$

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

$$y=e^{\frac{b^2}{2}+C},C\in \mathbb{R}$$

Solution

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Evaluate

$$e^{\frac{b^2}{2}+C}$$

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

$$e^C\times e^{\frac{b^2}{2}}$$

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

$$C{e}^{\frac{b^2}{2}}$$

$$y=C{e}^{\frac{b^2}{2}},C\in \mathbb{R}$$

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