Question

solve the linear equation x dy/dx + 3y = 6x its power is 3 solve this urgently

Answer 1

Answer on Question #38379 - Math - Differential Calculus

$$x \frac {d y}{d x} + 3 y = 6 x$$

Let's divide this equation by $x$:

$$\frac {d y}{d x} + 3 \frac {y}{x} = 6$$

Firstly let's find general solution of homogeneous equation

$$\frac {d y}{d x} + 3 \frac {y}{x} = 0$$

Let's solve this equation using separation of variables.

$$\frac {d y}{d x} = - \frac {3 y}{x}$$ $$\frac {d y}{y} = - \frac {3 d x}{x}$$

Integrating this equation:

$$\ln y = - 3 \ln x + \ln c$$ $$y = \frac {c}{x ^ {3}}$$

To solve non-homogeneous initial equation let's take constant $c$ as function of $x$: $c(x)$. Substituting $y = \frac{c(x)}{x^3}$ this into the equation we get:

$$x \cdot \frac {c ^ {\prime} \cdot x ^ {3} - c \cdot 3 x ^ {2}}{x ^ {6}} + \frac {3 c}{x ^ {3}} = 6 x$$ $$\frac {c ^ {\prime}}{x ^ {2}} = 6 x$$ $$c ^ {\prime} = 6 x ^ {3}$$ $$c = \frac {3}{2} x ^ {4} + c _ {1}$$

Thus general solution of initial equation is

$$y(x) = \frac{c(x)}{x^3} = \frac{3}{2}x + \frac{c}{x^3}$$

where $c \in \mathbb{R}$ is an arbitrary constant.