证明: 方程 [tex=4.857x1.5]3FFD+v/21Xhfm5t6bRuHxcweMtW1dz9ThTWatczAIZY=[/tex] 能在变换 [tex=2.429x1.0]VwrA7N9LmKT/YTXSmqh2iQ==[/tex] 之下化为可分离变量方程,并求方程 [tex=6.143x1.429]3FFD+v/21Xhfm5t6bRuHxetTE7HRfps0OqoY/WgcjF0=[/tex] 的通解.
证 原方程变形为 [tex=5.071x2.429]s4QumE5pv31BDaGg1MTOdMmd76tYrtW0fwJge6TFMLIOxJi3/7T7aTEG2SBtfhXZ[/tex], 因而应作代换 [tex=2.429x1.0]VwrA7N9LmKT/YTXSmqh2iQ==[/tex],有[tex=15.571x2.429]OfN5aOx06yWa+GdVBQgjRQ417UiLyf/UjXmxjNWSP1h8AIRxU0zW1RGLGaInpAEXuuVr97D84jHfHxrWEbx7SRZhEbuIE1H+gr0McpThjYpxAFUqpacdoXXEzfxoDy1kAsl7CajHSo4dAAMvo8YgVKEulweHw9vS1NYogxahJgA=[/tex]则原方程化为 [tex=7.857x2.429]WMyvC+x71zlq8XCeSTqSUKtvuJV7DwGg+HkNjdnhQE6vIzph4sghry9mkJlCi1BRT7SUwdeV9CAzAOYkWE+ZMshAIR9cOn8/PM0eiG++M3w=[/tex], 即 [tex=6.357x2.714]sFaNK9tspAWwmMYRxuIim5lmIyT9H1oGgN31RElIX6IHMUyyYdhSXYxfze+muanZ[/tex].显然上式为可分离变量方程,积分得[tex=14.857x2.714]C8mi5Vxh8s9vs9rG03eacsapqNvj/cFNhRAOD7Rl3VcqUTZCw7VyNt2UlCwJgZOFZxNkIcQGtTkYRdRCCWMFYg+NQ6q65fB7iXjnjfUtnwA=[/tex] .对上式求出积分后,将 [tex=2.429x1.0]VwrA7N9LmKT/YTXSmqh2iQ==[/tex] 代回即得其通解.由 [tex=5.5x1.429]kGeMKNtKwhIQJWsOKuiuC5Cm/5CzIGPKa5lR/o0i+KU=[/tex] 知 [tex=5.929x1.357]DGMQ6ORGEo1BKGo7vbEKiQ==[/tex] 即 [tex=4.929x1.357]aTOW8ZVmZelPCUt6+WsWqA==[/tex], 代入上式积分得[tex=22.571x2.714]C8mi5Vxh8s9vs9rG03eacj+jjt5OvWW9wTrqZ0hMypXcrUc+t0MfcZ0ShuUt7m7VQ4hwOOtw73dLRzWkXvh8fZeKooliDo/klHpghr6dpYA6BCV2XLvCrZMyiwtgmMVHu1OZg3iOAghrOSFrV0MYGg==[/tex] 即 [tex=23.786x1.571]pHWzkeLS0F+8UPZ31m3rfDrGxP8Lp/D6dz+bvAWL5m0xFqdKsv9CUUQBwMJbYDOSE80vlpBQDgTtpcolqz5yyA94zbWmZ5pmFzwLTNRFY+Z++7MZSmvNt2+/wIl7Ew0j24W0qD4mEelhHJ6Gw+FzuloWWm/20Z9RWtvt23t9jhIssBSDKtq14gwe5DM9krzv[/tex] 故 [tex=5.5x1.429]dzD5uDbsLfCBfT3hVUleCdtIRJIHAcNylAoCXiY4iyI=[/tex], 即 [tex=5.071x1.429]yooivv0N14FgZ9MWahhPAo5JsPiS06Ej3zkPWX8kZ/4=[/tex].
举一反三
- 用适合的变量代换将下列方程化为可分离变量的方程，然后求出通解: [tex=6.143x1.429]X/QCuE3cfKlDKT46qXDD6DDvLLfvoYJzv4Kyi2m75dI=[/tex]
- 证明方程 [tex=5.643x2.643]veMIbIHrCKyfJD6p8CsZieV/mC7jauoF+RoXvFL11rxcZNCHFWI1bp9PcV7QjXfuLz8jFJG3FjoRv6p+Zfkmnw==[/tex] 经变换 $x y=u$ 可化为变量分离方程,并由此求解方程：[tex=8.429x1.571]8HRcqzX3v4Y2lj/bxKtUWyTaeJGkmxPo/lnb2KrFyUkh3bTJjq7hgObaU0hI8NF68rCBoV64ntgfXyGigpHhLQ==[/tex]
- 求下列方程的通解:[tex=7.571x1.429]3FFD+v/21Xhfm5t6bRuHxWFuv4EpuF3VbaZZeMLOu5yV/mE0QbJzAqdKR0J92yGC/KBVynJdibPCIxwDZ05djQ==[/tex]
- 已知齐次方程 [tex=7.0x1.429]3FFD+v/21Xhfm5t6bRuHxTbB0Mag4GI6R3I7VVO3D46ILByZBIEP0JJ8QeJkV2pl[/tex] 的通解为 [tex=9.786x1.357]R8CaP2PSMTet3SxiACIdCqi71AuVuCWAoC2XVush6tQ=[/tex] 求非齐次线性方程[tex=7.071x1.429]3FFD+v/21Xhfm5t6bRuHxTbB0Mag4GI6R3I7VVO3D47tZ2hdiqsSAQjsFj1a/6su[/tex] 的通解。
- 求高阶方程 [tex=7.571x1.429]3FFD+v/21Xhfm5t6bRuHxTbB0Mag4GI6R3I7VVO3D44GhiNnMz35Fpdbc0O+de6i[/tex]
内容
- 0
用适当的变换将方程 [tex=6.929x1.429]fOaIfa35tG+Q4agdkW1dh1pfUQKtxon7abt6qxNh1PM=[/tex] 化为可分离变量的方程,并求出通解.
- 1
用适当的变换将下列方程化为可分离变量的方程,并求出通解[tex=5.929x1.5]By58ejDam90c98+I2RT93KlRqbjho8NvaRlhE2K7eu0=[/tex].
- 2
求方程[tex=7.643x1.429]3FFD+v/21Xhfm5t6bRuHxTbB0Mag4GI6R3I7VVO3D45qS9KH2nRLII7XwXjNAtW6[/tex]的一般解.
- 3
试具体解出分离变量和变量代换方程：[tex=5.357x1.429]3FFD+v/21Xhfm5t6bRuHxVUawmsBvJ/iGEZhI90Aqq8=[/tex]
- 4
用适当的变换将下列方程化为可分离变量的方程,并求出通解[tex=6.214x2.571]3je+BcHiLg705j+MffaTKlIz/LNtgAl/UYlLml0C4v8=[/tex].