3. 方程$x y' + xy = y $的通解为
A: \[y=\mathit{c}\,{{e}^{-x}}\]
B: \[y=\mathit{c}x\,{{e}^{-x}}\]
C: \[y=\mathit{c}x\,{{e}^{-x^2}}\]
D: \[y=\mathit{c}x^2\,{{e}^{-x}}\]
A: \[y=\mathit{c}\,{{e}^{-x}}\]
B: \[y=\mathit{c}x\,{{e}^{-x}}\]
C: \[y=\mathit{c}x\,{{e}^{-x^2}}\]
D: \[y=\mathit{c}x^2\,{{e}^{-x}}\]
举一反三
- 方程$(x^2+1)(y^2-1) + xy y' = 0$的通解为 A: $y^2 = C \frac{e^{-x^2}}{x^2}$ B: $y = C \frac{e^{-x^2}}{x^2}$ C: $y^2 = C \frac{e^{-x^2}}{x^2}+1$ D: $y=C \frac{e^{-x^2}}{x^2}+1$
- 设\(z = u{e^v}\)，\(u = {x^2} + {y^2}\)，\(v = xy\)，则\( { { \partial z} \over {\partial x}}=\) A: \({e^{xy}}({x^2}y + {y^3} + 2x)\) B: \({e^{xy}}({x}y^2 + {y^3} + 2x)\) C: \({e^{xy}}({x}y + {y^3} + 2x)\) D: \({e^{xy}}({x^2}y + {y^2} + 2x)\)
- 方程${{x}^{2}}{{y}^{''}}-(x+2)(x{{y}^{'}}-y)={{x}^{4}}$的通解是( ) A: $y={{C}_{1}}x+{{C}_{2}}{{e}^{x}}-(\frac{1}{2}{{x}^{3}}+{{x}^{2}})$ B: $y={{C}_{1}}x+{{C}_{2}}{{e}^{x}}-(\frac{1}{2}{{x}^{3}}+{{x}^{4}})$ C: $y={{C}_{1}}x+{{C}_{2}}x{{e}^{x}}-(\frac{1}{2}{{x}^{3}}+{{x}^{4}})$ D: $y={{C}_{1}}x+{{C}_{2}}x{{e}^{x}}-(\frac{1}{2}{{x}^{3}}+{{x}^{2}})$
- 设\(z = u{e^v}\)，\(u = {x^2} + {y^2}\)，\(v = xy\)，则\( { { \partial z} \over {\partial y}}=\)（ ）。 A: \({e^{xy}}({x}y^2 + {x^3} + 2y)\) B: \({e^{xy}}({x^2}y + {x^3} + 2y)\) C: \({e^{xy}}({x}y^2 + {x^3} + 2x)\) D: \({e^{xy}}({x}y+ {x^3} + 2y)\)
- 方程\( y' = {x^2}{y^2} \)的通解为( )。 A: \( y = {C \over { { x^3}}} \) B: \( y = { { - 3} \over { { x^3} + C}} \) C: \( y = C{x^3} \) D: \( y = C + {x^3} \)