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  • Show that the given differential equation is homogeneous and solve each of them. y' equals x plus y by x

Show that the given differential equation is homogeneous
and solve each of them.

    Q2.    y' = \frac{x+y}{x}

Answers (1)

best_answer

the above differential eq can be written as,

\frac{dy}{dx} = F(x,y)=\frac{x+y}{x}............................(i)

Now, F(\lambda x,\lambda y)=\frac{\lambda x+\lambda y}{\lambda x} = \lambda ^{0}F(x,y)
Thus the given differential eq is a homogeneous equation
Now, to solve the substitute y = vx
Differentiating on both sides wrt x
\frac{dy}{dx}= v +x\frac{dv}{dx}
                                 
Substitute this value in equation (i)

v+x\frac{dv}{dx}= \frac{x+vx}{x} = 1+v
\\x\frac{dv}{dx}= 1\\ dv = \frac{dx}{x}
Integrating on both sides, we get; (and substitute the value of v =\frac{y}{x})

\\v =\log x+C\\ \frac{y}{x}=\log x+C\\ y = x\log x +Cx
this is the required solution 

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manish

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