solving the initial value problem by separation of variables dy/dx=(4(sqrt of y)lnx))/x, y(e)=9
To solve the initial value problem using separation of variables, we need to follow these steps:
Step 1: Write the differential equation in the form dy/dx = g(x) * f(y).
The given differential equation is dy/dx = (4√y * ln(x))/x. We can rewrite it as:
dy/dx = (4√y * ln(x))/x
Step 2: Rearrange the equation to separate the variables x and y.
Multiply both sides of the equation by x and divide both sides by 4√y:
√y * dy = (ln(x) / x) * dx
Step 3: Integrate both sides of the equation.
Integrating both sides of the equation, we get:
∫√y dy = ∫(ln(x) / x) dx
To integrate the left side, we can use the power rule for integration:
(2/3) * y^(3/2) = ∫(ln(x) / x) dx
Step 4: Find the indefinite integral of (∫ln(x) / x) dx.
To find the indefinite integral of (∫ln(x) / x) dx, we can use integration by parts:
Let u = ln(x) and dv = dx.
Then du = (1/x) dx and v = x.
Using the integration by parts formula: ∫u dv = uv - ∫v du, we can calculate:
∫(ln(x) / x) dx = ln(x) * x - ∫x * (1/x) dx
= xln(x) - ∫dx
= xln(x) - x + C
Step 5: Substitute back into the equation and solve for y.
Now, we substitute the integration result back into the original equation:
(2/3) * y^(3/2) = xln(x) - x + C
To solve for y, we need to isolate it. Multiply both sides by (3/2) and then raise both sides to the power of 2/3:
y^(3/2) = (3/2) * (xln(x) - x + C)
y = [(3/2) * (xln(x) - x + C)]^(2/3)
Finally, we can substitute the initial condition y(e) = 9 to find the value of C:
9 = [(3/2) * (e * ln(e) - e + C)]^(2/3)
Simplify the equation and solve for C. Then substitute the found value of C back into the equation to get the solution for the initial value problem.