Solve the differential equation

Evaluate

$$t^{\prime }=w$$

Rewrite the expression

$$\frac{dt}{dw}=w$$

Transform the expression

$$dt=wdw$$

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

$$\int 1dt=\int wdw$$

Calculate

More Steps

Evaluate

$$\int 1dt$$

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

$$t$$

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

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

$$t+C_1=\int wdw,C_1\in \mathbb{R}$$

Calculate

More Steps

Evaluate

$$\int wdw$$

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

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

Add the numbers

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

Add the numbers

$$\frac{w^2}{2}$$

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

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

$$t+C_1=\frac{w^2}{2}+C_2,C_1\in \mathbb{R},C_2\in \mathbb{R}$$

Solution

$$t=\frac{w^2}{2}+C,C\in \mathbb{R}$$

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