The equation3y′=3xy+2x2+2y2x2, (∗)can be written in the form y′=f(y/x), i.e., it is homogeneous, so we can use the substitution
u=y/x to obtain a separable equation with dependent variable u=u(x). Introducing this substitution and using the fact that y′=xu′+u we can write (∗) asy′=xu′+u=f(u)where f(u)=Separating variables we can write the equation in the formg(u)du=dxxwhere g(u)= . An implicit general solution with dependent variable u can be written in the formln(x)−=CTransforming u=y/x back into the variables x and y and using the initial condition y(1)=1 we findC=Finally solve for y to obtain the explicit solution of the initial value problemy=
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3 because Negative numbers equal less than 0. See the number line attached below. In the number line, all the negative numbers are on the left of the zero and all the positive numbers are on the right of the 0.
