Let y = y(x) be the solution curve of the differential equation x*(dy/dx) - y = sqrt(y² + 16*x²), with initial condition y(1) = 3. Find y(2).

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Accepted Answer

15. Rewrite: x*dy/dx - y = sqrt(y² + 16x²). Let v = y/x, y = vx, dy/dx = v + x*dv/dx. Then x*(v + x*dv/dx) - vx = x*sqrt(v²+16). So x²*(dv/dx) = x*sqrt(v²+16) => x*(dv/dx) = sqrt(v²+16). Separable: dv/sqrt(v²+16) = dx/x. Integrate: ln|v + sqrt(v²+16)| = ln|x| + C. At x=1, y=3, v=3: ln(3 + sqrt(9+16)) = ln(3+5) = ln(8) = 0 + C => C = ln(8). So ln|v+sqrt(v²+16)| = ln(8x). At x=2: v+sqrt(v²+16)=16. Let u=sqrt(v²+16): v+u=16 => u=16-v => u²=v²+16 => (16-v)²=v²+16 => 256-32v+v²=v²+16 => 240=32v => v=7.5. y=vx=7.5*2=15.

Let y = y(x) be the solution curve of the differential equation x(dy/dx) - y = sqrt(y² + 16x²), with initial condition y(1) = 3. Find y(2).

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