Question (1 point) Match the second order linear equations with the Wronskian of (one of) their fundamental solution sets. 1. \( y^{\prime \prime}-\frac{1}{t} y^{\prime}-9 y=0, \quad t>0 \) 2. \( y^{\prime \prime}-\ln (t) y^{\prime}-9 y=0 \) 3. \( y^{\prime \prime}-\cos (t) y^{\prime}-9 y=0 \) 4. \( y^{\prime \prime}-4 y^{\prime}-9 y=0 \) 5. \( y^{\prime \prime}-2 t (1 point) Match the second order linear equations with the Wronskian of (one of) their fundamental solution sets. 1. y′′−t1y′−9y=0,t>0 2. y′′−ln(t)y′−9y=0 3. y′′−cos(t)y′−9y=0 4. y′′−4y′−9y=0 5. y′′−2ty′−9y=0 A. W(t)=7t B. W(t)=e4t c. W(t)=etln(t)−t D. W(t)=esin(t) E. W(t)=3exp(t2)

WXDUZD The Asker · Advanced Mathematics

Transcribed Image Text: (1 point) Match the second order linear equations with the Wronskian of (one of) their fundamental solution sets. 1. y′′−t1y′−9y=0,t>0 2. y′′−ln(t)y′−9y=0 3. y′′−cos(t)y′−9y=0 4. y′′−4y′−9y=0 5. y′′−2ty′−9y=0 A. W(t)=7t B. W(t)=e4t c. W(t)=etln(t)−t D. W(t)=esin(t) E. W(t)=3exp(t2)

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Sol.for\begin{array}{l}y^{\prime \prime}+a(x) y^{\prime}+b \int x \mid=0, \\w=c e^{\int-a(x) d x}, \quad \text { c is ConsH. }\end{array}C is Const.(1)\begin{array}{l}y^{\prime \prime}-\frac{1}{t} y^{\prime}+9 y=0 \\W=c e^{\int \frac{1}{t} d t}=c e^{\log t}=c t \\W(t)=7 t\end{array}(II)\begin{array}{c}y^{\prime \prime}-\ln (t) y^{\prime}-9 y=0 \\W=c e^{\int \ln (t) d t}=c e^{t \ln t-t} \\w(t)=e^{t \ln t-t}\end{array}(III)\begin{array}{c}y^{\prime \prime}-\cos (t) y^{\prime}-9 y=0 \\w=c e^{\int+\cos t d t}=c e^{\sin t} \\\omega(t)=e^{\sin (t)}\end{array}(iv)\begin{array}{l}y^{\prime \prime}-4 y^{\prime}-9 y=0 \\w(t)=c e^{\int 4 i d t}=c e^{4 t} \\\omega(t)=e^{4 t}\end{array}(V)\begin{array}{l}y^{\prime \prime}-2 t y^{\prime}-9 y=0 \\\omega(t)=c e^{\int 2 t d t}=c e^{\frac{q t^{2}}{2}}=c e^{t^{2}} \\\Rightarrow \quad \omega(t)=3 e^{t^{2}} \\\end{array} ...