Deferential Solutions for Various Equations

Running head: DEFERENTIAL SOLUTIONS
Deferential Solutions
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DEFERENTIAL SOLUTIONS
Question 1: Find the explicit general solution of xdx - (y^2)dy=0.
Xdx - (y^2)dy = 0
Or, xdx = (y^2)dy
integrating both sides we get,
(x^2)/2 + C1 = (y^3)/3 + C2
Or, 3(x^2) + 3C1 - 2(y^3) - 2C2 = 0
Let, 3C1 - 2C2 = C’
Therefore,
3(x^2) - 2(y^3) + C’ = 0
Hence, this is the general solution of the equation.
Question 2: Find the explicit general solution of 2xyy’ – y^2 +x^2 = 0 (Homogeneous
Substitution Method)
2xyy’ – y^2 + x^2 = 0
Or, dy/dx = (x^2 – y^2)
By the substitution method,
Let v = y/x
Therefore, y = vx

DEFERENTIAL SOLUTIONS
Or, dy/dx = v + x(dv/dx)
Therefore, by substituting the value we get,
v + x(dv/dx) = {x^2 – (v^2)(x^2)}/2vx^2
Or, dx/x = 2dv/((3v^2)-1)
Integrating both sides we get,
Logx + C1 = 1/3log ((3v^2)-1) + C2
Or, x^3= 3v^2 – 1 + C
Where C is the composite constant
Putting the value of v in the equation, we get,
x^3= {3(y^2)/(x^2)} – 1 + C
Question 3: Find the explicit general solution of xy’ + y + 4 =0.
a) Integrating Factor
xdy/dx + y + 4 = 0
Or, dy/dx + y/x = -4/x
By integrating factor method we get,
P = 1/x
Therefore the integrating factor = e^∫dx/x = e^log x = x