Given the ODE (yx + y2) dx - x2 dy = 0 Write the equation in the form y\' = f(y/

Question

Given the ODE (yx + y2) dx - x2 dy = 0 Write the equation in the form y' = f(y/x) = Next use the substitution u = yjx to write the equation as an ODE with independent variable x and dependent variable u, i.e., in the form u' = f(u) - u/x = Notice that this equation is separable. If we separate the variables we obtain two separate integrals (Do not add an arbitrary constant as we already put one there.)): du = +C and dx = +C Converting back to the original variables .r and y leads to an implicit general solution which can be written as ln(x)+ = C

Explanation / Answer

y' = yx+y^2 /x^2

u=y/x => du/dx = (xdy/dx - y)/x^2 = (dy/dx- y/x)/x = dy/dx -u /x = u^2 /x

du/u^2 = dx/x => int (du/u^2 ) = -1/u +c1 and int (dx/x) = ln x + c2

=>-x/y = lnx + c

=> lnx + x/y +c =0

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.