(dydx)=−y2+x22xyandy(1)=4 Please, solve the differential equation. Write the method you used and solve for the dependent variable it it is possible.

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(dydx)=−y2+x22xyandy(1)=4  Please, solve the differential equation. Write the method you used and solve for the dependent variable it it is possible.

doplovif · Accepted Answer

The given equation:y′=−y2+x22xy,y(1)=4The given ODE can be written as:y′=−y2x−x2yimplies y′+(12x)y=(−x2)y−1 (1)A first oredr Bernoulli ODE has the form ofy′+p(x)y=q(x)ynIn this case we have also given a first order Bernoulli ODE withp(x)=12x,q(x)=−x2,n=−1Note that the general solution is obtained by substituting υ=y1−n and solving 11−nυ′+p(x)υ=q(x)Putting υ=y2 in (1) and using υ′=2yy′ we getυ′+(1x)=−xNow by multipling the integrating factor exp⁡(∫1xdx)=exp⁡(ln⁡x)=x we getυ=−x3+c13xNow substitute back υ=y2y2=−x3+c13xApply initial conditions y(1)=4 gives us c1=49 and so we get y2=−x3+493xNote that y=−−x3+493x, not possible as it does not gives us y(1)=4So, the required solution is y=+−x3+493x

(dy/dx) = -(y^2 + x^2) / (2xy) and y(1) = 4 Write the method you used and solve for the dependent variable it it is possible

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