Exercise · ZW-EXACT-EQUATION-005 · ★☆☆ Basic
The equation \((2x+y)\,dx+(x+2y)\,dy=0\) is exact. Which function \(F\) satisfies \(F_x=M\) and \(F_y=N\)?
Choose one option.
Show answer
- C. \(F=x^2+xy+y^2\)
Explanation
Integrating \(M\) with respect to \(x\) gives \(x^2+xy+h(y)\), and \(F_y=x+h'(y)=x+2y\) gives \(h=y^2\). The term \(xy\) must appear only once.
Review the concept: Exact equation
An exact equation is a differential equation \(M\,dx+N\,dy=0\) whose left side is the total differential of a function \(F(x,y)\).
\[\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}\]
More practice problems: exact equation
- ZW-EXACT-EQUATION-001 Which of these equations is exact?
- ZW-EXACT-EQUATION-002 If M=xy^2, what is (∂ M)/(∂ y)?
- ZW-EXACT-EQUATION-003 If N=x^2y+sin x, what is (∂ N)/(∂ x)?
- ZW-EXACT-EQUATION-004 For which value of k is (kxy+3) dx+(x^2+y) dy=0 exact?
- ZW-EXACT-EQUATION-006 Solve the exact equation (2xy+3) dx+(x^2-1) dy=0.
- ZW-EXACT-EQUATION-007 Solve (2x+e^y) dx+xe^y dy=0.
- ZW-EXACT-EQUATION-008 The equation 2xy dx+(x^2+1) dy=0 has solution x^2y+y=C. If y(0)=2, what is y(1)?
- ZW-EXACT-EQUATION-009 The equation y dx+2x dy=0 is not exact. Which function should it be multiplied by to make it exact?