Cross-Multiply to Simplify: We are given the differential equation (dy)/(dx) = (5x)/(y). To simplify this equation, we can cross-multiply to get rid of the fraction. So, we multiply both sides by y and dx to get: y * (dy) = 5x * (dx)Integrate Both Sides: Now we have a separable differential equation. We can integrate both sides to find the relationship between y and x. Integrate y * (dy) with respect to y and 5x * (dx) with respect to x. ∫y dy = ∫5x dxPerform Integration: Perform the integration on both sides. The integral of y with respect to y is (1/2)y^2. The integral of 5x with respect to x is (5/2)x^2. So, we have: (1/2)y^2 = (5/2)x^2 + C, where C is the constant of integration.Multiply by 2: To express the relationship between y and x, we can multiply the entire equation by 2 to get rid of the fraction. 2 * (1/2)y^2 = 2 * (5/2)x^2 + 2C This simplifies to: y^2 = 5x^2 + 2CSimplify Constant: We can leave the constant as 2C or simply rename it as a new constant, let's call it C' (since the constant of integration can be any real number). So, the simplified form of the differential equation is: y^2 = 5x^2 + C'

Algebra 2

Simplify rational expressions

Full solution

\frac{dy}{dx} = \frac{5x}{y}

Cross-Multiply to Simplify: We are given the differential equation $\frac{dy}{dx} = \frac{5x}{y}$ . To simplify this equation, we can cross-multiply to get rid of the fraction. $\newline$ So, we multiply both sides by $y$ and $dx$ to get: $\newline$ $y \cdot dy = 5x \cdot dx$
Integrate Both Sides: Now we have a separable differential equation. We can integrate both sides to find the relationship between $y$ and $x$ . Integrate $y \cdot \text{d}y$ with respect to $y$ and $5x \cdot \text{d}x$ with respect to $x$ . $\int y \, \text{d}y = \int 5x \, \text{d}x$
Perform Integration: Perform the integration on both sides. $\newline$ The integral of $y$ with respect to $y$ is $(1/2)y^2$ . $\newline$ The integral of $5x$ with respect to $x$ is $(5/2)x^2$ . $\newline$ So, we have: $\newline$ $(1/2)y^2 = (5/2)x^2 + C$ , where $C$ is the constant of integration.
Multiply by $2$ : To express the relationship between $y$ and $x$ , we can multiply the entire equation by $2$ to get rid of the fraction. $\newline$ $2 \times \left(\frac{1}{2}\right)y^2 = 2 \times \left(\frac{5}{2}\right)x^2 + 2C$ $\newline$ This simplifies to: $\newline$ $y^2 = 5x^2 + 2C$
Simplify Constant: We can leave the constant as $2C$ or simply rename it as a new constant, let's call it $C'$ (since the constant of integration can be any real number). $\newline$ So, the simplified form of the differential equation is: $\newline$ $y^2 = 5x^2 + C$