Question

21) Show that the following differential equation is homogenous and solve it 2xy+y^2-2x^2.dy/dx

Answer 1

dear student

$$\begin{gathered} 2xy+y^2-2x^2\frac{dy}{dx}=0 \\ \frac{dy}{dx}=\frac{2xy+y^2}{2x^2} \\ puttingy=vx \\ \frac{dy}{dx}=v+x\frac{dv}{dx} \\ \frac{dy}{dx}=v+x\frac{dv}{dx}=\frac{2xy+y^2}{2x^2}=\frac{2v+v^2}{2} \\ AfirstorderDifferentialEquationisHomogeneouswhenitcanbeinthisform: \\ \frac{dy}{dx}=f\left(\frac{y}{x}\right)=f\left(v\right) \\ hencehomogeneous \\ v+x\frac{dv}{dx}=\frac{2v+v^2}{2} \\ \frac{dv}{v^2}=\frac{dx}{2x} \\ \frac{-2}{v}=\log x+c \\ \frac{-2x}{y}=\log x+c \end{gathered}$$

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