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1.2 First-order linear differential equations

category Differential Equations 2020. 5. 12. 23:12
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1.2 First-order linear differential equations

 

1) Definition of general first-order linear differential equation

\[\frac{dy}{dt}+a(t)y=b(t)\]

 

nonhomogeneous first-order linear differential equation for b(t)≠0

 

e.g.

\[\frac{dy}{dt}=y^{2}+sin(t)\]

\[\frac{dy}{dt}=cos(y)+t\]

are nonlinear equations for y^2 and cos(y)

 

 

2) Definition of homogeneous first-order linear differential equation

\[\frac{dy}{dt}+a(t)y=0\]

 

general solution of the homogeneous equation

\[\frac{\frac{dy}{dt}}{y}=-a(t)\]

\[\frac{\frac{dy}{dt}}{y}\equiv \frac{d}{dt}ln\left | y(t) \right |\]

\[\frac{d}{dt}ln\left | y(t) \right |=-a(t)\]

\[ln\left | y(t) \right |=-\int a(t)dt+c_{1}\]

\[\left | y(t) \right |=e^{\left ( -\int a(t)dt+c_{1} \right )}=ce^\left ( -\int a(t)dt \right)\]

\[\left | y(t) e^\left ( \int a(t)dt \right)\right |=c\]

\[y(t)=c e^\left ( -\int a(t)dt \right)\]

 

with initial-value

\[\frac{dy}{dt}+a(t)y=0\]\[\frac{dy}{dt}+a(t)y=0, \, y(t_{0})=y_{0}\]

\[\frac{\frac{dy}{dt}}{y}=-a(t)\]

\[\int_{t_{0}}^{t}\frac{d}{ds}ln\left | y(s) \right |ds=-\int_{t_{0}}^{t}a(s)ds\]

\[ln\left | y(t) \right |-ln\left | y(t_{0}) \right |=ln\left | \frac{y(t)}{y(t_{0})} \right |=-\int_{t_{0}}^{t}a(s)ds\]

\[\left | \frac{y(t)}{y(t_{0})} \right |=e^{-\int_{t_{0}}^{t}a(s)ds}\]

\[\left | \frac{y(t)}{y(t_{0})} e^{\int_{t_{0}}^{t}a(s)ds}\right |=1\]

\[\frac{y(t)}{y(t_{0})} e^{\int_{t_{0}}^{t}a(s)ds}=1 \,\, or \,\, \frac{y(t)}{y(t_{0})} e^{\int_{t_{0}}^{t}a(s)ds}=-1\]

\[when\, t=t_{0}\]

\[\frac{y(t_{0})}{y(t_{0})} e^{\int_{t_{0}}^{t_{0}}a(s)ds}=1e^{0}=1\]

\[therefore\, \frac{y(t)}{y(t_{0})} e^{\int_{t_{0}}^{t}a(s)ds}=1\]

\[y(t)=y(t_{0})e^{-\int_{t_{0}}^{t}a(s)ds}=y_{0}e^{-\int_{t_{0}}^{t}a(s)ds}\]

 

 

3) nonhomogeneous equation

\[\frac{dy}{dt}+a(t)y=b(t)\]

\[\mu (t)\frac{dy}{dt}+\mu (t)a(t)y=\mu (t)b(t)\]

\[\frac{d}{dt}\left ( \mu (t)y\right )=\mu (t)\frac{dy}{dt}+\frac{d\mu}{dt}y\]
\[if \,\, \frac{d\mu(t)}{dt}=\mu(t)a(t),\]

\[then\,\,\frac{d}{dt}\left ( \mu (t)y\right )=\mu (t)\frac{dy}{dt}+\mu (t)a(t)y\]

\[and \, \, \frac{d}{dt}\left ( \mu (t)y\right )=\mu (t)b(t)\]

\[\frac{d\mu(t)}{dt}=\mu(t)a(t)\]

\[\frac{d\mu(t)}{\mu(t)}=a(t) dt\]

\[ln\left | \mu (t) \right |=\int a(t)dt\]

\[\mu(t)=e^{\int a(t)dt}\]

 

\[\frac{d}{dt}\left ( \mu (t)y\right )=\mu (t)b(t)\]

\[\mu (t)y=\int \mu (t)b(t)dt+c\]

\[y=\frac{1}{\mu (t)}\left ( \int \mu (t)b(t)dt+c \right )\]

\[y=e^{-\int a(t)dt}\left ( \int e^{\int a(t)dt}b(t)dt+c \right )\]

 

with initial condition

\[y(t_{0})=y_{0}\]

\[\frac{dy}{dt}+a(t)y=b(t)\]

 

\[\frac{d}{dt}\left ( \mu (t)y\right )=\mu (t)b(t)\]

\[\mu (t)y- \mu (t_{0})y_{0}=\int_{t_{0}}^{t} \mu (s)b(s)ds\]

\[y= \frac{1}{\mu (t)}\left ( \mu (t_{0})y_{0}+\int_{t_{0}}^{t} \mu (s)b(s)ds\right )\]

 

 

 

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