Question

求微分方程的解

dy(t)/dt=-50[y(t)-cos(t)], 请问y的解是多少？如果y(0)=0呢

Answer 1

Answer 2

dy/dt=-50(y-cost)=-50y+50cost
先求对应的齐次方程dy/dt=-50y
dy/y=-50dt
ln|y|=-50t+C
即y=C e^(-50t)
由常数变易法，令y=C(t)e^(-50t)
代入原方程得C'(t)=50cost e^(50t)
注：令U=∫cost e^(50t)dt
C(t)=50U=∫cost d [e^(50t)]=cost e^(50t)+∫sint e^(50t)dt
=cost e^(50t)+1/50 ∫sint d[e^(50t)]
=cost e^(50t)+1/50 sint e^(50t) - 1/50 ∫cost e^(50t)
=cost e^(50t)+1/50 sint e^(50t) - U/50
得U=50/2501 cost e^(50t)+1/2501 sint e^(50t) +C1
故C(t)=2500/2501 cost e^(50t)+50/2501 sint e^(50t) +C
所以原方程的通解为
y=2500/2501 cost +50/2501 sint +Ce^(-50t)
y(0)=2500/2501 +C=0，C=-2500/2501
故y(0)=0时的特解为y=2500/2501 cost +50/2501 sint - 2500/2501 e^(-50t)

Answer 3

搞清楚对谁求啊。【GM/2（x-R）】t^2+C1t+C2.其中C1,C2为常数。