User:Egm6321.f12.team4.harris/Homework3 R*3.1

Homework 3 Problem R*3.1 – Proof that only one integration constant is necessary in the solution of a general, non-homogeneous, linear, first order differential equation.
sec12-2

Statement
Consider equation 3.1.1 as a general, non-homogeneous, linear, first order ordinary differential equation with varying coefficients (L1-ODE-VC).
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$$ P(x)y'+Q(x)y=R(x) $$     (3.1.1)
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It can be shown that only one integration constant is necessary when solving the L1-ODE-VC via Euler's Integrating Factor Method. The proof can be seen below.

Given
If $$ P(x)\ne0 $$, then dividing equation 3.1.1 by $$ P(x) $$ would yield
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$$ a_1(x)y'+a_0(x)y=b(x) $$      (3.1.2)
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such that $$ a_1(x)=1 $$, $$ a_0(x)=\frac{Q(x)}{P(x)} $$, and $$ b(x)=\frac{R(x)}{P(x)}$$. To solve this L1-ODE-VC, one can employ Euler's Integrating Factor Method by finding an equation $$ h(x) $$ such that
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$$ h(x)=exp(\int^xa_0(s)ds + k1) $$      (3.1.3) and
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$$ y(x)=\frac{1}{h(x)}[\int^xh(s)b(s)ds+k_2] $$      (3.1.4) where $$ s $$ is a dummy integration variable.
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Solution
To begin proving that the integration constant $$ k_1 $$ is not necessary to solve equation 3.1.1, one can rewrite
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$$ exp(\int^xa_0(s)ds+k_1)=exp(k_1)exp(\int^xa_0(s)ds) $$      (3.1.5)
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One can then redefine $$ k_1 $$ as $$ k_1=exp(k_1) $$. Substituting the right hand side of equation 3.1.5 into equation 3.1.4, it is shown that
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$$ y(x)=\frac{1}{k_1exp(\int^xa_0(s)ds)}[\int^x[k_1exp\int^xa_0(s)ds]b(s)ds+k_2] $$
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Since $$ k_1 $$ is a constant, it can be brought out of the integral.
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$$ y(x)=\frac{1}{k_1exp(\int^xa_0(s)ds)}[k_1\int^x[exp\int^xa_0(s)ds]b(s)ds+k_2] $$
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Canceling the $$ k_1 $$ term in the denominator of the left hand term with the right hand term, one is left with
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$$ y(x)=\frac{1}{exp(\int^xa_0(s)ds)}[\int^x[exp\int^xa_0(s)ds]b(s)ds+\frac{k_2}{k_1}] $$      (3.1.6) One can then redefine the integration constant in equation 3.1.6 to be
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$$ k_2=\frac{k_2}{k_1} $$
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Thus equation 3.1.6 becomes
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$$ y(x)=\frac{1}{exp(\int^xa_0(s)ds)}[\int^x[exp\int^xa_0(s)ds]b(s)ds+k_2] $$
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This proves that only one integration constant, $$ k_2 $$, is necessary for the solution of equation 3.1.1 via Euler's Integration Factor Method.
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Author and References

 * Solved and Typed by -- Kaitlin Harris