Exercise · ZW-EXACT-EQUATION-007 · ★★☆ Intermediate
Solve \((2x+e^{y})\,dx+xe^{y}\,dy=0\).
Choose one option.
Show answer
- D. \(x^2+xe^{y}=C\)
Explanation
It is exact because \(M_y=e^{y}=N_x\). From \(\int M\,dx=x^2+xe^{y}\) and \(F_y=xe^{y}=N\), no extra function \(h(y)\) remains.
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-005 The equation (2x+y) dx+(x+2y) dy=0 is exact. Which function F satisfies F_x=M and F_y=N?
- ZW-EXACT-EQUATION-006 Solve the exact equation (2xy+3) dx+(x^2-1) 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?