Let y(x) be the solution of the differential equation 2(x² + x^(5/4)) dy - y*(x + x^(1/4)) dx = 2*x^(9/4) dx for x 0, passing through the point (1, 1 - (4/3)*ln 2). Find y(16).

Question

Let y(x) be the solution of the differential equation 2(x² + x^(5/4)) dy - y*(x + x^(1/4)) dx = 2*x^(9/4) dx for x > 0, passing through the point (1, 1 - (4/3)*ln 2). Find y(16).

Accepted Answer

4*(31/3 - (8/3)*ln 3). Rewriting as a linear first-order ODE and computing the integrating factor by simplifying the coefficient of y leads to a solvable integral; applying the initial condition determines the constant, and substituting x=16 gives the answer.

Let y(x) be the solution of the differential equation 2(x² + x^(5/4)) dy - y(x + x^(1/4)) dx = 2x^(9/4) dx for x > 0, passing through the point (1, 1 - (4/3)*ln 2). Find y(16).

Solution

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