91Ó°ÊÓ

If the right-hand side of the equation$\frac{dy}{dx}=f\left(x,y\right)$ can be expressed as a function of the ratio $\frac{y}{x}$alone, then we say the equation is homogeneous.

Equations of the form$\frac{dy}{dx}=G\left(ax+by\right)$

When the right-hand side of the equation$\frac{dy}{dx}=f\left(x,y\right)$ can be expressed as a function of the combination $ax+by$, where a and b are constants, that is,$\frac{dy}{dx}=G\left(ax+by\right)$then the substitution$z=ax+by$ transforms the equation into a separable one.

A first-order equation that can be written in the form $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)y^n$, where P(x) and Q(x) are continuous on an interval (a, b) and n is a real number, is called a Bernoulli equation.

We have used various substitutions for y to transform the original equation into a new equation that we could solve. In some cases, we must transform both x and y into new variables, say u and v. This is the situation for equations with linear coefficients-that is, equations of the form

.$\left(a_{1}x+b_{1}y+c_1\right)dx+\left(a_{2}x+b_{2}y+c_2\right)dy=0$

Given,

By Evaluating,

$\begin{array}{rcl}\mathrm{θ}dy-yd\mathrm{θ}& =& \sqrt{\mathrm{θ}yd\mathrm{θ}}\\ \mathrm{θ}dy& =& \sqrt{\mathrm{θ}yd\mathrm{θ}}+yd\mathrm{θ}\\ & =& \left(\sqrt{\mathrm{θ}y}+y\right)d\mathrm{θ}\\ \frac{dy}{d\mathrm{θ}}& =& \frac{\sqrt{\mathrm{θ}y}+y}{\mathrm{θ}}\\ & =& \frac{\sqrt{y}}{\sqrt{\mathrm{θ}}}+\frac{y}{\mathrm{θ}}\\ & =& \sqrt{\frac{y}{\mathrm{θ}}+}\frac{y}{\mathrm{θ}}\end{array}$

Let us consider $v=\frac{y}{\mathrm{θ}}$. Then,

$\begin{array}{rcl}\sqrt{\frac{y}{\mathrm{θ}}}+\frac{y}{\mathrm{θ}}& =& G\left(\frac{Y}{\mathrm{θ}}\right)\\ & =& G\left(v\right)\\ & =& \sqrt{v}+v\end{array}$

So, the given equation is homogeneous.

Now rewritten the same equation,

$$\begin{gathered} \frac{dy}{d\mathrm{θ}}=\frac{y}{\mathrm{θ}}+\sqrt{\frac{y}{\mathrm{θ}}} \\ \frac{dy}{d\mathrm{θ}}+\left(-\frac{1}{\mathrm{θ}}\right)y=\left(\frac{1}{\sqrt{\mathrm{θ}}}\right)y^{\frac{1}{2}} \\ \frac{dy}{d\mathrm{θ}}+P\left(\mathrm{θ}\right)y=Q\left(\mathrm{θ}\right)y^n \end{gathered}$$

It seems that the given equation is Bernoulli.

Therefore, the given equation is the form of homogeneous and Bernoulli equation.