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(x(dy)/(dx) -y ) sin (y /x) =x^(2)e^(x) ...

`(x(dy)/(dx) -y ) sin (y /x) =x^(2)e^(x) , y = vx `

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To solve the differential equation \((x \frac{dy}{dx} - y) \sin\left(\frac{y}{x}\right) = x^2 e^x\) using the substitution \(y = vx\), we will follow these steps: ### Step 1: Substitute \(y = vx\) We start by substituting \(y = vx\), where \(v\) is a function of \(x\). This gives us: \[ \frac{dy}{dx} = v + x \frac{dv}{dx} \] ...
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Solution of the differential equation (x(dy)/(dx)-y)sin((y)/(x))=x^(2)e^(x) is

(dy)/(dx)=(y)/(x)+x sin((y)/(x))

(dy) / (dx) = (sin x) / (e ^ (y))

sin x(dy)/(dx)+y=y^(2)

(dy) / (dx) = (y) / (x) + sin ((y) / (x))

(dy)/(dx)=e^(-y)*sin x+e^((x-y))

(dy)/(dx)=e^(-y)sin x+e^((x-y))

(dy/dx)+(y/x)=sin x^(2)

(dy)/(dx)=(x^2+e^(x))/(y)

using the subsitiution y = vx : (1) (x(dy)/(dx) -y)^(e^(y/x) =x^(2) cos x