6x² + 2y² y' (+) xy can be written in the form y'= f(y/z), i.e., it is homogeneous, so we can use the substitution u=y/a

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6x 2y Y Xy Can Be Written In The Form Y F Y Z I E It Is Homogeneous So We Can Use The Substitution U Y A 1 (47.33 KiB) Viewed 11 times

6x² + 2y² y' (+) xy can be written in the form y'= f(y/z), i.e., it is homogeneous, so we can use the substitution u=y/a to obtain a separable equation with dependent variable u = u(x). Introducing this substitution and using the fact that y'= zu' + u we can write (+) as y = xu' +u = f(u) where f(u) Separating variables we can write the equation in the form g(u) du where g(u) u/(6+u^2) An implicit general solution with dependent variable u can be written in the form ‒‒ P ■ (6+2u^2)/u (x^2)/(3x^4) = 1 we find Transforming = y/ back into the variables a and y and using the initial condition y(1) C= 7/6 dz a Finally solve for y to obtain the explicit solution of the initial value problem y=xsqrt(7x^2-6) OLD C. W