Question

solve : dy/dx + xy= x​

Answer 1

Step-by-step explanation:

Homogeneous Differential Equations

A first order Differential Equation is Homogeneous when it can be in this form:

dydx = F( yx )

We can solve it using Separation of Variables but first we create a new variable v = yx

v = yx which is also y = vx

And dydx = d (vx)dx = v dxdx + x dvdx (by the Product Rule)

Which can be simplified to dydx = v + x dvdx

Using y = vx and dydx = v + x dvdx we can solve the Differential Equation.

An example will show how it is all done:

Example: Solve dydx = x2 + y2xy

Can we get it in F( yx ) style?

Start with: x2 + y2xy

Separate terms: x2xy + y2xy

Simplify: xy + yx

Reciprocal of first term: ( yx )−1 + yx

Yes, we have a function of (y/x).

So let's go:

Start with: dydx = ( yx )−1 + yx

y = vx and dydx = v + x dvdx : v + x dvdx = v−1 + v

Subtract v from both sides: x dvdx = v−1

Now use Separation of Variables:

Separate the variables: v dv = 1x dx

Put the integral sign in front: ∫v dv = ∫ 1x dx

Integrate: v22 = ln(x) + C

Then we make C = ln(k): v22 = ln(x) + ln(k)

Combine ln: v22 = ln(kx)

Simplify: v = ±√(2 ln(kx))

Now substitute back v = yx

Substitute v = yx: yx = ±√(2 ln(kx))

Simplify: y = ±x √(2 ln(kx))

And we have the solution.

The positive portion looks like this:

y = x sqrt(2 ln(kx))

Another example:

Example: Solve dydx = y(x−y)x2

Can we get it in F( yx ) style?

Start with: y(x−y)x2

Separate terms: xyx2 − y2x2

Simplify: yx − ( yx )2

Start with: dydx = 1−y/x1+y/x

y = vx and dydx = v + x dvdx v + x dvdx = 1−v1+v

Subtract v from both sides: x dvdx = 1−v1+v − v

Then: x dvdx = 1−v1+v − v+v21+v

Simplify: x dvdx = 1−2v−v21+v

Now use Separation of Variables:

Separate the variables: 1+v1−2v−v2 dv = 1x dx

Put the integral sign in front: ∫ 1+v1−2v−v2 dv = ∫ 1x dx

Integrate: − 12 ln(1−2v−v2) = ln(x) + C

Then we make C = ln(k): − 12 ln(1−2v−v2) = ln(x) + ln(k)

Combine ln: (1−2v−v2)−½ = kx

Combine ln: (1−2v−v2)−½ = kx

Square and Reciprocal: 1−2v−v2 = 1k2x2

Now substitute back v = yx

Substitute v = yx: 1−2( yx )−( yx )2 = 1k2x2

Multiply through by x2: x2−2xy−y2 = 1k2Change signs: y2+2xy−x2 = − 1k2

Replace − 1k2 by c: y2+2xy−x2 = c

Add 2x2 to both sides: y2+2xy+x2 = 2x2+c

Factor: (y+x)2 = 2x2+c

Square root: y+x = ±√(2x2+c)

Subtract x from both sides: y = ±√(2x2+c) − x

And we have the solution.

Answer 2

Step-by-step explanation:

above answer is correct