Solve the differential equation

Evaluate

$$y^{\prime }\times 2y=\cos \left(t\right)$$

Rewrite the expression

$$2y\times y^{\prime }=\cos \left(t\right)$$

Rewrite the expression

$$2y\frac{dy}{dt}=\cos \left(t\right)$$

Transform the expression

$$2ydy=\cos \left(t\right)dt$$

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

$$\int 2ydy=\int \cos \left(t\right)dt$$

Calculate

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Evaluate

$$\int 2ydy$$

$$\text{Use the property of integral }\int kf(x)dx=k\int f(x)dx$$

$$2\times \int ydy$$

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

$$2\times \frac{y^{1+1}}{1+1}$$

Simplify

More Steps

Evaluate

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

Add the numbers

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

Add the numbers

$$\frac{y^2}{2}$$

$$2\times \frac{y^2}{2}$$

$$\text{Cancel out the common factor }2$$

$$1\times y^2$$

Multiply the terms

$$y^2$$

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

$$y^2+C_1,C_1\in \mathbb{R}$$

$$y^2+C_1=\int \cos \left(t\right)dt,C_1\in \mathbb{R}$$

Calculate

More Steps

Evaluate

$$\int \cos \left(t\right)dt$$

$$\text{Use the property of integral }\int \cos (x)dx=\sin (x)$$

$$\sin \left(t\right)$$

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

$$\sin \left(t\right)+C_2,C_2\in \mathbb{R}$$

$$y^2+C_1=\sin \left(t\right)+C_2,C_1\in \mathbb{R},C_2\in \mathbb{R}$$

Solution

$$y^2=\sin \left(t\right)+C,C\in \mathbb{R}$$

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