Question

Obtain the solution of the equation.
xy'=4y.

Answer 1

Answer on Question #82190 – Math – Differential Equations

Question

Obtain the solution of the equation.

$$x y ^ {\prime} = 4 y.$$

Solution

$$x y ^ {\prime} = 4 y;$$

In this ODE we can do a separation of variables.

$$\begin{array}{l} x \frac {d y}{d x} = 4 y; \\ \frac {d y}{d x} = \frac {4 y}{x}; \\ \frac {d y}{y} = 4 \frac {d x}{x}; \\ \int \frac {d y}{y} = 4 \int \frac {d x}{x}; \\ \ln y = 4 \ln x + A; \\ \end{array}$$

where $A$ is a some constant. We can rewrite it:

$$A = 4 \ln B;$$

where $B$ is another constant. Therefore, we have

$$\ln y = 4 \ln x + 4 \ln B = 4 \ln (B x) = \ln (B x) ^ {4} = \ln (C x ^ {4});$$

where $C = B^4$

Finally,

$$y (x) = C x ^ {4}.$$

Answer: $y = C x^4$

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