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$$4\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x} = 3u$$ , $$u(x,0)=4e^{-x}-e^{-5x}$$
let $$ U =X(x)T(t) $$
so
$$4X\frac{\partial T}{\partial t}+T\frac{\partial X}{\partial x} = 3XT$$
$$4\frac{\partial T}{T \partial t}+\frac{\partial X}{X \partial x} = 3$$
$$\left( 4\frac{\partial T}{T \partial t}-3 \right) +\frac{\partial X}{X \partial x} = 0 $$
let K = constant
$$\frac{\partial X}{X \partial x} =\left( 3 - 4\frac{\partial T}{T \partial t} \right) = k $$
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$$\frac{\partial X}{X \partial x} = k $$
$$\frac{d X}{X} = k dx$$
$$ X = C_1e^{kx}$$
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$$\left( 3 - 4\frac{\partial T}{T \partial t} \right) = k $$
$$ 4\frac{\partial T}{T \partial t}= 3 - k $$
$$ \frac{\partial T}{T \partial t}= \frac{1}{4}(3 - k) $$
$$ \frac{d T}{T}= \frac{1}{4}(3 - k) dt $$
$$ T = C_2e^{\frac{1}{4}(3 - k) t} $$
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general solution
$$ u(x,t) = C e^{kx}e^{\frac{1}{4}(3 - k) t} $$ then $$ C=C_1C_2 $$
$$ u(x,0) = C e^{kx} = 4e^{-x}-e^{-5x} $$ <<How do I solve this equation?
let $$ U =X(x)T(t) $$
so
$$4X\frac{\partial T}{\partial t}+T\frac{\partial X}{\partial x} = 3XT$$
$$4\frac{\partial T}{T \partial t}+\frac{\partial X}{X \partial x} = 3$$
$$\left( 4\frac{\partial T}{T \partial t}-3 \right) +\frac{\partial X}{X \partial x} = 0 $$
let K = constant
$$\frac{\partial X}{X \partial x} =\left( 3 - 4\frac{\partial T}{T \partial t} \right) = k $$
_________________________________________________________________________________
$$\frac{\partial X}{X \partial x} = k $$
$$\frac{d X}{X} = k dx$$
$$ X = C_1e^{kx}$$
_________________________________________________________________________________
$$\left( 3 - 4\frac{\partial T}{T \partial t} \right) = k $$
$$ 4\frac{\partial T}{T \partial t}= 3 - k $$
$$ \frac{\partial T}{T \partial t}= \frac{1}{4}(3 - k) $$
$$ \frac{d T}{T}= \frac{1}{4}(3 - k) dt $$
$$ T = C_2e^{\frac{1}{4}(3 - k) t} $$
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general solution
$$ u(x,t) = C e^{kx}e^{\frac{1}{4}(3 - k) t} $$ then $$ C=C_1C_2 $$
$$ u(x,0) = C e^{kx} = 4e^{-x}-e^{-5x} $$ <<How do I solve this equation?