1.

dy/dx - 3y/2x = 2x/y\(\frac{dy}{dx}-\frac{3y}{2x}=\frac{2x}{y}\)

Answer»

\(\frac{dy}{dx}-\frac{3y}{2x}=\frac{2x}{y}\)

⇒ \(\frac{dy}{dx}=\frac{2x}{y}+\frac{3}{2}\frac{y}{x}\)

⇒ \(\cfrac{dy}{dx}=\cfrac{2}{\frac yx}+\frac{3}{2}\frac{y}{x}\)   ----(i)

let y = vx

then \(\frac{dy}{dx}=V+x\frac{dv}{dx}\)

∴ v + \(x\frac{dv}{dx}=\frac{2}{v}+\frac{3}{2}v\)  ----(from i)

⇒ \(x\frac{dv}{dx}=\frac{2}{v}+\frac{3}{2}v-v=\frac{2}{v}+\frac{1}{2}v=\frac{4+v^2}{2v}\)

⇒ \(\frac{2v}{4+v^2}\,dv=\frac{1}{x}dx\)

⇒\(\int\frac{2v}{4+v^2}\,dv=\int\, \frac{1}{x}\,dx\)

⇒ log | 4 + v2 | = logx + log c

⇒ log | 4 + v2 | = log cx

⇒ 4+v2 = cx

⇒ 4 + \((\frac{y}{x})^2=cx\)      \(∵v=\frac{y}{x}\)

⇒ 4x2 + y2 = cx3  

which is solution of given differential equation.


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