求微分方程的通解例题解

来源：蜘蛛抓取(WebSpider) 时间：2016-03-01 10:57 标签： 微分方程求通解
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数学实验：求微分方程的解
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你可能喜欢大一高数:求以下微分方程的通解(高手进)y′ - y=x^2y″ = yy′ =e^(y/x) +(y/x)麻烦告诉我这些是什么类型的方程,是齐次方程,二阶常系数齐次线性微分方程,像这类的.答得好一定采纳!_百度作业帮
大一高数:求以下微分方程的通解(高手进)y′ - y=x^2y″ = yy′ =e^(y/x) +(y/x)麻烦告诉我这些是什么类型的方程,是齐次方程,二阶常系数齐次线性微分方程,像这类的.答得好一定采纳!
1.一阶常系数线性非齐次方程齐次通解为y=e^x特解设为y=ax平方＋bx+cy'=2ax+b2ax+b-ax平方-bx-c=x^2-ax^2+(2a-b)x+b-c=x^2-a=12a-b=0b-c=0所以a=-1b=-2c=-2特解为y=-x^2-2x-2通解为y=ce^x-x^2-2x-22.二阶常系数齐次线性方程r平方=1r1=1,r2=-1通解为y=c1e^x+c2e^(-x)3.齐次方程令y/x=uy=uxy'=xu'+u代入原式,得xu'+u=e^u+uxdu/dx=e^u-e^(-u)du=-1/xdx两边积分,得e^(-u)=-lnx+lnce^(-u)=lnc/xc/x=e^[e^(-u)]c=xe^[e^(-u)]即通解为xe^[e^(-y/x)]=c
请问你是怎么判断出第一个是非齐次方程的？要经过计算吗？
这个是齐次的
y′ - y=x^2
是非齐次的，主要看等号右边是0还是不是0.
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