Question
Mathematics Question on Differential equations
For x∈R, let y(x) be the solution of the differential equation (x^2 -5)$$\frac{dy}{dx}$$-2xy = - 2x$$(x^2-5)^2 such that y(2)=7. Then the maximum value of the function y(x) is
Answer
Given:
(x^2 -5)$$\frac{dy}{dx}-2xy = - 2x(x2−5)2
Rearrange the equation to solve for dxdy:
dxdy=x2−52x(x2−5)2−2xy
Now, we have a first-order separable differential equation. Let's separate variables:
2x(x2−5)2−2xydy=x2−5dx
Next, integrate both sides:
∫2x(x2−5)2−2xydy=∫x2−5dx
Integrating these expressions will provide us with the function y(x).