Question

Solve the second order DE [x^2y^{\prime\prime}+3xy^\prime=2x\]

Answer 1

\[x^2y^{\prime\prime}+3xy^\prime=2x\]
\[\text {let}\quad \ln x=t \]\[x=e^t\]\[\frac{\text d x}{\text dt}=e^t\]
\[e^{2t}y^{\prime\prime}+3e^ty^\prime=2e^t\]\[y^{\prime\prime}+(3-1)y^\prime=2e^t\]\[y^{\prime\prime}+2y^\prime=2e^t\]

Answer 2

\[y^{\prime\prime}_c+2y_c^\prime=0\]
\[m^2+2m=0\]\[m(m+2)=0\]\[m=0,-2\]
\[y_c=A+Be^{-2t}\]

Answer 3

\[y_p^{\prime\prime}+2y_p^\prime=2e^t\]
\[y_p=Ce^t;\qquad y_p^\prime=Ce^t;\qquad y_p^{\prime\prime}=Ce^t\]
\[Ce^t+2Ce^t=2e^t\]\[3Ce^t=2e^t\]\[C=\frac 23\]\[y_p=\frac23e^t\]

Answer 4

\[y(t)=A+Be^{-2t}+\frac 23e^t\]
\[y(x)=A+Bx^{-2}+\frac 23x\]

Answer 5

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