I was able to successfully complete a and b in this question but I am very confused by c. How can I prove a general solution when I need to add both together? I am confused because c is requiring the addition of problems a and b but that seems nonsensical as b is already proving c without addition.
Linear Algebra and Differential Equations
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linear-algebra
integration
ordinary-differential-equations
derivatives
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0How did you show $y_c(x)$ is a general solution to the DE in part (a)? – 2017-02-02
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0I used an integrating factor that allowed integration of both sides and solved for y. – 2017-02-02
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0That's actually the easiest problem of the three. Sub $y_c + y_p$ into the left side of the DE, and show that it will turn into the DE with $y_c$ + the DE with $y_p$, thus equaling the right side. This is the principal of superposition, and any linear DE has this property. – 2017-02-02
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0Thank you, it's making more sense. For superposition, is it left in a differential form? I'm having a bit of trouble understanding what the question wants. – 2017-02-02
1 Answers
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We assume two facts: $$\frac{d(y_c)}{dx} + Py_c = 0$$ $$\frac{d(y_p)}{dx} + Py_p = Q$$ This result follows: $$\frac{d(y_c + y_p)}{dx} + P(y_c + y_p)= [\frac{d(y_c)}{dx} + Py_c]+[\frac{d(y_p)}{dx} + Py_p]=0 + Q = Q$$
This is, in general, known as superposition. Let $L(y)$ be a linear differential equation of $y$. Then if $L(y_1)=A(x)$, and $L(y_2)=B(x)$, then $L(y_1+y_2)= A(x) + B(x)$.
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0Oh, That makes so much sense! Thank you so much! – 2017-02-02