Solution Found!
Let and y2(x):=x. Verify that L[y1](x) = x sinx and
Chapter 6, Problem 23E(choose chapter or problem)
Let \(L[y]:=y^{\prime \prime}+y^{\prime}+x y, y_{1}(x):=\sin x\) , and \(y_{2}(x):=x\). Verify that \(L\left[y_{1}\right](x)=x \sin x\) and \(L\left[y_{2}\right](x)=x^{2}+1\). Then use the superposition principle (linearity) to find a solution to the differential equation:
(a) \(L[y]=2 x \sin x-x^{2}-1\).
(b) \(L[y]=4 x^{2}+4-6 x \sin x\).
Equation Transcription:
Text Transcription:
L[y]:=y'''+y'+xy, y1(x):=sin x
y_2(x):=x
L[y_1](x)=x sin x
L[y_2](x)=x^2+1
L[y]=2x sin x-x^2-1
L[y]=4x^2+4-6x sin x
Questions & Answers
QUESTION:
Let \(L[y]:=y^{\prime \prime}+y^{\prime}+x y, y_{1}(x):=\sin x\) , and \(y_{2}(x):=x\). Verify that \(L\left[y_{1}\right](x)=x \sin x\) and \(L\left[y_{2}\right](x)=x^{2}+1\). Then use the superposition principle (linearity) to find a solution to the differential equation:
(a) \(L[y]=2 x \sin x-x^{2}-1\).
(b) \(L[y]=4 x^{2}+4-6 x \sin x\).
Equation Transcription:
Text Transcription:
L[y]:=y'''+y'+xy, y1(x):=sin x
y_2(x):=x
L[y_1](x)=x sin x
L[y_2](x)=x^2+1
L[y]=2x sin x-x^2-1
L[y]=4x^2+4-6x sin x
ANSWER:
Solution:
Step 1:
In this problem we need to verify that and
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