Find general solution
[t - sqrt(ty) ] dy/dt = y
To find the general solution of the given differential equation, we can follow these steps:
Step 1: Rearrange the equation to separate the variables.
[t - sqrt(ty)] dy = y dt
Step 2: Move all terms involving dy to one side and all terms involving dt to the other side.
[t - sqrt(ty)] dy - y dt = 0
Step 3: Find a common factor in the left side of the equation.
dy[t - sqrt(ty)/dt] - y = 0
Step 4: Apply the quotient rule to simplify the left side.
(dy/dt)[t - sqrt(ty)] - y = 0
Step 5: Multiply through by dt to get the derivative terms on one side and the variables on the other side.
dy[t - sqrt(ty)] - y dt = 0
Step 6: Distribute the dy term.
t dy - sqrt(ty) dy - y dt = 0
Step 7: Rearrange terms while grouping y and dy terms together.
t dy - y dt = sqrt(ty) dy
Step 8: Divide through by y*sqrt(t) to isolate dy on the left side.
dy/dt - (y/t) dt = 1/sqrt(t) dy
Step 9: Recognize that the left side of the equation is in the form (dy/dt + P(t)y) dt = Q(t) dy, where P(t) = -1/t and Q(t) = 1/sqrt(t).
Step 10: Now, we can use the integrating factor method to solve the equation. The integrating factor, denoted by IF, is given by the formula IF = e^(integral P(t) dt). Let's calculate it.
IF = e^(integral -1/t dt)
= e^(-ln|t|)
= 1/t
Step 11: Multiply the entire equation by the integrating factor.
(1/t)(dy/dt) - (1/t)(y/t) dt = (1/t) (1/sqrt(t)) dy
dy/t - y/t^2 dt = 1/(t^3/2) dy
Step 12: Recognize that the left side of the equation is the derivative of (y/t). Integrate both sides accordingly.
d(y/t) = 1/(t^3/2) dy
Integrate both sides:
∫ d(y/t) = ∫ 1/(t^3/2) dy
y/t = ∫ 1/(t^3/2) dy
Step 13: Integrate the right side.
y/t = 2√t + C
Step 14: Multiply through by t to isolate y.
y = 2t√t + Ct
Therefore, the general solution of the given differential equation is y = 2t√t + Ct, where C is the constant of integration.
[t-√(ty)] dy/dt = y
let y = tv. Then you should be able to separate the variables