Find the solution to y^{\prime \prime }+y^{\prime }-2y=0;y\left ( 0\right ) =1,y^{\prime }\left ( 0\right ) =1 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving\begin{align*} r^{2}+r-2 & =0\\ \left ( r+2\right ) \left ( r-1\right ) & =0 \end{align*}
Hence the roots are r_{1}=-2,r_{2}=1. Roots are real and distinct. The two solutions are\begin{align*} y_{1} & =e^{-2t}\\ y_{2} & =e^{t} \end{align*}
The general solution is linear combination of the above two solutions\begin{align*} y & =c_{1}y_{1}+c_{2}y_{2}\\ & =c_{1}e^{-2t}+c_{2}e^{t} \end{align*}
Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( 0\right ) =1) to the general solution gives\begin{equation} 1=c_{1}+c_{2} \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =-2c_{1}e^{-2t}+c_{2}e^{t}. Applying second initial condition to this results in\begin{equation} 1=-2c_{1}+c_{2} \tag{2} \end{equation} Equation (1,2) are now solved for c_{1},c_{2}. From (1), c_{1}=1-c_{2}. Substituting this into (2) gives \begin{align*} 1 & =-2\left ( 1-c_{2}\right ) +c_{2}\\ & =-2+2c_{2}+c_{2}\\ & =-2+3c_{2} \end{align*}
Hence c_{2}=\frac{1+2}{3}=1. Therefore c_{1}=1-1=0. Hence\begin{align*} c_{1} & =0\\ c_{2} & =1 \end{align*}
Substituting these back into the general solution gives y\left ( t\right ) =e^{t} Since the solution is exponential, it will grow in time and blows up. Here is sketch of the solution.
Find the solution to y^{\prime \prime }+4y^{\prime }+3y=0;y\left ( 0\right ) =2,y^{\prime }\left ( 0\right ) =-1 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving\begin{align*} r^{2}+4r+3 & =0\\ \left ( r+3\right ) \left ( r+1\right ) & =0 \end{align*}
Hence the roots are r_{1}=-3,r_{2}=-1. Roots are real and distinct. The two solutions are\begin{align*} y_{1} & =e^{-3t}\\ y_{2} & =e^{-t} \end{align*}
The general solution is linear combination of the above two solutions\begin{align*} y & =c_{1}y_{1}+c_{2}y_{2}\\ & =c_{1}e^{-3t}+c_{2}e^{-t} \end{align*}
Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( 0\right ) =2) to the general solution gives\begin{equation} 2=c_{1}+c_{2} \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =-3c_{1}e^{-3t}-c_{2}e^{-t}. Applying second initial condition to this results in\begin{equation} -1=-3c_{1}-c_{2} \tag{2} \end{equation} Equation (1,2) are now solved for c_{1},c_{2}. From (1), c_{1}=2-c_{2}. Substituting this into (2) gives \begin{align*} -1 & =-3\left ( 2-c_{2}\right ) -c_{2}\\ & =-6+3c_{2}-c_{2}\\ & =-6+2c_{2} \end{align*}
Hence c_{2}=\frac{-1+6}{2}=2.5. Therefore c_{1}=2-2.5=0.5. Hence\begin{align*} c_{1} & =0.5\\ c_{2} & =2.5 \end{align*}
Substituting these back into the general solution gives y\left ( t\right ) =0.5e^{-3t}+2.5e^{-t} At t becomes large, both solutions decay to zero. So we expect the general solution to go to zero very fast. Here is a sketch.
Find the solution to 6y^{\prime \prime }-5y^{\prime }+y=0;y\left ( 0\right ) =4,y^{\prime }\left ( 0\right ) =0 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving 6r^{2}-5r+1=0 Hence r_{1,2}=\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}, where \Delta =b^{2}-4ac=25-\left ( 4\right ) \left ( 6\right ) =1. Since \Delta >0, the roots will be real and distinct. The roots are\begin{align*} r_{1,2} & =\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}\\ & =\frac{5}{12}\pm \frac{1}{12} \end{align*}
Hence the roots are r_{1}=\frac{1}{2},r_{2}=\frac{1}{3}. Roots are real and distinct. The two solutions are\begin{align*} y_{1} & =e^{\frac{1}{2}t}\\ y_{2} & =e^{\frac{1}{3}t} \end{align*}
The general solution is linear combination of the above two solutions\begin{align*} y & =c_{1}y_{1}+c_{2}y_{2}\\ & =c_{1}e^{\frac{1}{2}t}+c_{2}e^{\frac{1}{3}t} \end{align*}
Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( 0\right ) =4) to the general solution gives\begin{equation} 4=c_{1}+c_{2} \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =\frac{1}{2}c_{1}e^{\frac{1}{2}t}+\frac{1}{3}c_{2}e^{\frac{1}{3}t}. Applying second initial condition to this results in\begin{equation} 0=\frac{1}{2}c_{1}+\frac{1}{3}c_{2} \tag{2} \end{equation} Equation (1,2) are now solved for c_{1},c_{2}. From (1), c_{1}=4-c_{2}. Substituting this into (2) gives \begin{align*} 0 & =\frac{1}{2}\left ( 4-c_{2}\right ) +\frac{1}{3}c_{2}\\ & =2-\frac{1}{2}c_{2}+\frac{1}{3}c_{2}\\ & =2-\frac{1}{6}c_{2} \end{align*}
Hence c_{2}=12. Therefore c_{1}=4-12=-8. Hence\begin{align*} c_{1} & =-8\\ c_{2} & =12 \end{align*}
Substituting these back into the general solution gives y\left ( t\right ) =-8e^{\frac{1}{2}t}+12e^{\frac{1}{3}t} Since e^{\frac{1}{2}t} grows faster than e^{\frac{1}{3}t} and since e^{\frac{1}{2}t} has negative coefficient, then the solution will go to -\infty as t increases. Here is sketch of the solution
Find the solution to y^{\prime \prime }+3y^{\prime }=0;y\left ( 0\right ) =-2,y^{\prime }\left ( 0\right ) =3 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving\begin{align*} r^{2}+3r & =0\\ r\left ( r+3\right ) & =0 \end{align*}
Hence the roots are r_{1}=0,r_{2}=-3. Roots are real and distinct. The two solutions are\begin{align*} y_{1} & =1\\ y_{2} & =e^{-3t} \end{align*}
The general solution is linear combination of the above two solutions y=c_{1}+c_{2}e^{-3t} Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( 0\right ) =-2) to the general solution gives\begin{equation} -2=c_{1}+c_{2} \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =-3c_{2}e^{-3t}. Applying second initial condition to this results in\begin{equation} 3=-3c_{2} \tag{2} \end{equation} Hence c_{2}=-1. Therefore c_{1}=-1. Substituting these back into the general solution gives y\left ( t\right ) =-1-e^{-3t} As t\rightarrow \infty , the term e^{-3t}\rightarrow 0 and we are left with -1. Hence \lim _{t-\infty }y\left ( t\right ) =-1. Here is sketch of the solution
Find the solution to y^{\prime \prime }+5y^{\prime }+3y=0;y\left ( 0\right ) =1,y^{\prime }\left ( 0\right ) =0 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving r^{2}+5r+3=0 Hence r_{1,2}=\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}, where \Delta =b^{2}-4ac=25-\left ( 4\right ) \left ( 3\right ) =13. Since \Delta >0, the roots will be real and distinct. The roots are\begin{align*} r_{1,2} & =\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}\\ & =\frac{-5}{2}\pm \frac{\sqrt{13}}{2} \end{align*}
Hence the roots are r_{1}=\frac{-5}{2}+\frac{\sqrt{13}}{2},r_{2}=\frac{-5}{2}-\frac{\sqrt{13}}{2}. The two solutions are\begin{align*} y_{1} & =e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}\\ y_{2} & =e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t} \end{align*}
The general solution is linear combination of the above two solutions y=c_{1}e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+c_{2}e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t} Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( 0\right ) =1) to the general solution gives\begin{equation} 1=c_{1}+c_{2} \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =c_{1}\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+c_{2}\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t} Applying second initial condition to this results in\begin{equation} 0=c_{1}\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) +c_{2}\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) \tag{2} \end{equation} From (1), c_{1}=1-c_{2} and from (2) \begin{align*} 0 & =\left ( 1-c_{2}\right ) \left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) +c_{2}\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) \\ & =\frac{-5}{2}+\frac{\sqrt{13}}{2}+\frac{5}{2}c_{2}-\frac{\sqrt{13}}{2}c_{2}-\frac{5}{2}c_{2}-\frac{\sqrt{13}}{2}c_{2}\\ & =-\frac{5}{2}+\frac{\sqrt{13}}{2}-\sqrt{13}c_{2}\\ c_{2} & =\frac{-5}{2\sqrt{13}}+\frac{1}{2}\\ & =\frac{-5+\sqrt{13}}{2\sqrt{13}} \end{align*}
Therefore c_{1}=1+\frac{5-\sqrt{13}}{2\sqrt{13}} and the solution becomes\begin{align*} y\left ( t\right ) & =\left ( 1+\frac{5-\sqrt{13}}{2\sqrt{13}}\right ) e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+\left ( \frac{-5+\sqrt{13}}{2\sqrt{13}}\right ) e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}\\ & =e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+\frac{5-\sqrt{13}}{2\sqrt{13}}e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+\left ( \frac{-5+\sqrt{13}}{2\sqrt{13}}\right ) e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}\\ & =e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+\frac{5\sqrt{13}-13}{26}e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+\left ( \frac{-5\sqrt{13}+13}{26}\right ) e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}\\ & =\frac{1}{26}\left ( 26e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+\left ( 5\sqrt{13}-13\right ) e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+\left ( -5\sqrt{13}+13\right ) e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}\right ) \\ & =\frac{1}{26}\left ( 26e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+5\sqrt{13}e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}-13e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}-5\sqrt{13}e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}+13e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}\right ) \\ & =\frac{1}{26}\left ( 13e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}+5\sqrt{13}e^{\left ( \frac{-5}{2}+\frac{\sqrt{13}}{2}\right ) t}-5\sqrt{13}e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}+13e^{\left ( \frac{-5}{2}-\frac{\sqrt{13}}{2}\right ) t}\right ) \end{align*}
Here is sketch of the solution showing that y\rightarrow 0 as t\rightarrow \infty
Find the solution to 2y^{\prime \prime }+y^{\prime }-4y=0;y\left ( 0\right ) =0,y^{\prime }\left ( 0\right ) =1 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving 2r^{2}+r-4=0 Hence r_{1,2}=\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}, where \Delta =b^{2}-4ac=1-\left ( 4\right ) \left ( 2\right ) \left ( -4\right ) =33. Since \Delta >0, the roots will be real and distinct. The roots are\begin{align*} r_{1,2} & =\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}\\ & =\frac{-1}{4}\pm \frac{\sqrt{33}}{4} \end{align*}
Hence the roots are r_{1}=\frac{1}{4}+\frac{\sqrt{33}}{4},r_{2}=\frac{1}{4}-\frac{\sqrt{33}}{4}. The two solutions are\begin{align*} y_{1} & =e^{\left ( -\frac{1}{4}+\frac{\sqrt{33}}{4}\right ) t}\\ y_{2} & =e^{\left ( -\frac{1}{4}-\frac{\sqrt{33}}{4}\right ) t} \end{align*}
The general solution is linear combination of the above two solutions y=c_{1}e^{\left ( -\frac{1}{4}+\frac{\sqrt{33}}{4}\right ) t}+c_{2}e^{\left ( -\frac{1}{4}-\frac{\sqrt{33}}{4}\right ) t} Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( 0\right ) =0) to the general solution gives\begin{equation} 0=c_{1}+c_{2} \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =c_{1}\left ( -\frac{1}{4}+\frac{\sqrt{33}}{4}\right ) e^{\left ( \frac{1}{4}+\frac{\sqrt{33}}{4}\right ) t}+c_{2}\left ( -\frac{1}{4}-\frac{\sqrt{33}}{4}\right ) e^{\left ( \frac{1}{4}-\frac{\sqrt{33}}{4}\right ) t} Applying second initial condition to this results in\begin{equation} 1=c_{1}\left ( -\frac{1}{4}+\frac{\sqrt{33}}{4}\right ) +c_{2}\left ( -\frac{1}{4}-\frac{\sqrt{33}}{4}\right ) \tag{2} \end{equation} From (1), c_{1}=-c_{2} and from (2) \begin{align*} 1 & =-c_{2}\left ( -\frac{1}{4}+\frac{\sqrt{33}}{4}\right ) +c_{2}\left ( -\frac{1}{4}-\frac{\sqrt{33}}{4}\right ) \\ & =\frac{1}{4}c_{2}-\frac{\sqrt{33}}{4}c_{2}-\frac{1}{4}c_{2}-\frac{\sqrt{33}}{4}c_{2}\\ & =\frac{-\sqrt{33}}{2}c_{2}\\ c_{2} & =\frac{-2}{\sqrt{33}} \end{align*}
Therefore c_{1}=\frac{2}{\sqrt{33}} and the solution becomes y=\frac{2}{\sqrt{33}}e^{\left ( -\frac{1}{4}+\frac{\sqrt{33}}{4}\right ) t}-\frac{2}{\sqrt{33}}e^{\left ( -\frac{1}{4}-\frac{\sqrt{33}}{4}\right ) t} Since -\frac{1}{4}+\frac{\sqrt{33}}{4}=1.186 and -\frac{1}{4}-\frac{\sqrt{33}}{4}=-1.686 then the above can be written as y=\frac{2}{\sqrt{33}}e^{1.186t}-\frac{2}{\sqrt{33}}e^{-1.186t} Then we see that as t\rightarrow \infty the second term e^{-1.186t}\rightarrow 0 and we are left with e^{1.186t} which will go to \infty for large t. Hence \lim _{t\rightarrow \infty }y\left ( t\right ) =\infty Here is sketch of the solution
Find the solution to y^{\prime \prime }+8y^{\prime }-9y=0;y\left ( 1\right ) =1,y^{\prime }\left ( 1\right ) =0 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving\begin{align*} r^{2}+8r-9 & =0\\ \left ( r-1\right ) \left ( r+9\right ) & =0 \end{align*}
Hence the roots are r_{1}=1,r_{2}=-9. The two solutions are\begin{align*} y_{1} & =e^{t}\\ y_{2} & =e^{-9t} \end{align*}
The general solution is linear combination of the above two solutions y=c_{1}e^{t}+c_{2}e^{-9t} Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( 1\right ) =1) to the general solution gives\begin{equation} 1=c_{1}e^{1}+c_{2}e^{-9} \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =c_{1}e^{t}-9c_{2}e^{-9t} Applying second initial condition to this results in\begin{equation} 0=c_{1}e^{1}-9c_{2}e^{-9} \tag{2} \end{equation} From (1), c_{1}=\frac{1-c_{2}e^{-9}}{e^{1}}=e^{-1}-c_{2}e^{-10} and from (2) \begin{align*} 0 & =\left ( e^{-1}-c_{2}e^{-10}\right ) e^{1}-9c_{2}e^{-9}\\ & =1-c_{2}e^{-9}-9c_{2}e^{-9}\\ & =1+c_{2}\left ( -e^{-9}-9e^{-9}\right ) \\ 0 & =1+c_{2}\left ( -10e^{-9}\right ) \end{align*}
Hence c_{2}=\frac{1}{10}e^{9} Therefore c_{1}=e^{-1}-c_{2}e^{-10}=e^{-1}-\frac{1}{10}e^{9}e^{-10}=e^{-1}-\frac{1}{10}e^{-1}=\frac{9}{10}e^{-1} and the solution becomes\begin{align*} y & =\frac{9}{10}e^{-1}e^{t}+\frac{1}{10}e^{9}e^{-9t}\\ & =\frac{9}{10}e^{t-1}+\frac{1}{10}e^{9-9t} \end{align*}
Then we see that as t\rightarrow \infty the second term e^{9-9t}\rightarrow 0 and we are left with e^{t-1} which will go to \infty for large t. Hence \lim _{t\rightarrow \infty }y\left ( t\right ) =\infty Here is sketch of the solution.
Find the solution to 4y^{\prime \prime }-y=0;y\left ( -2\right ) =1,y^{\prime }\left ( -2\right ) =-1 and sketch the solution and describe its behavior as t increases.
solution
The characteristic equation is found by substituting y=e^{rt} into the ODE and simplifying, giving 4r^{2}-1=0 Hence the roots are r_{1}=\pm \frac{1}{2}. The two solutions are\begin{align*} y_{1} & =e^{\frac{1}{2}t}\\ y_{2} & =e^{-\frac{1}{2}t} \end{align*}
The general solution is linear combination of the above two solutions y=c_{1}e^{\frac{1}{2}t}+c_{2}e^{-\frac{1}{2}t} Now c_{1},c_{2} are found from initial conditions. Applying first initial condition (y\left ( -2\right ) =1) to the general solution gives\begin{equation} 1=c_{1}e^{-1}+c_{2}e \tag{1} \end{equation} Taking time derivative of the general solution gives y^{\prime }\left ( t\right ) =\frac{1}{2}c_{1}e^{\frac{1}{2}t}-\frac{1}{2}c_{2}e^{-\frac{1}{2}t} Applying second initial condition to this results in\begin{equation} -1=\frac{1}{2}c_{1}e^{-1}-\frac{1}{2}c_{2}e \tag{2} \end{equation} From (1), c_{1}=\frac{1-c_{2}e}{e^{-1}}=e-c_{2}e^{2} and from (2) \begin{align*} -1 & =\frac{1}{2}\left ( e-c_{2}e^{2}\right ) e^{-1}-\frac{1}{2}c_{2}e\\ & =\frac{1}{2}-\frac{1}{2}c_{2}e-\frac{1}{2}c_{2}e\\ & =\frac{1}{2}-c_{2}e \end{align*}
Hence c_{2}=\frac{1}{2}e^{-1}+e^{-1}=\frac{3}{2}e^{-1} Therefore c_{1}=e-\left ( \frac{3}{2}e^{-1}\right ) e^{2}=e-\frac{3}{2}e=-\frac{1}{2}e and the solution becomes\begin{align*} y & =c_{1}e^{\frac{1}{2}t}+c_{2}e^{-\frac{1}{2}t}\\ & =-\frac{1}{2}ee^{\frac{1}{2}t}+\frac{3}{2}e^{-1}e^{\frac{-1}{2}t}\\ & =-\frac{1}{2}e^{1+\frac{t}{2}}+\frac{3}{2}e^{-1-\frac{t}{2}} \end{align*}
Then we see that as t\rightarrow \infty the second term e^{-1-\frac{t}{2}}\rightarrow 0 and we are left with -\frac{1}{2}e^{1+\frac{t}{2}} which will go to -\infty for large t. Hence \lim _{t\rightarrow \infty }y\left ( t\right ) =-\infty Here is sketch of the solution.
Find the Wronskian of the given pair of functions e^{2t},e^{-\frac{3t}{2}}
solution
We are given y_{1}\left ( t\right ) =e^{2t},y_{2}\left ( t\right ) =e^{\frac{-3}{2}t}, hence by definition, the Wronskian is\begin{align*} W & =\begin{vmatrix} y_{1}\left ( t\right ) & y_{2}\left ( t\right ) \\ y_{1}^{\prime }\left ( t\right ) & y_{2}^{\prime }\left ( t\right ) \end{vmatrix} \\ & =\begin{vmatrix} e^{2t} & e^{-\frac{3t}{2}}\\ 2e^{2t} & -\frac{2}{3}e^{-\frac{3t}{2}}\end{vmatrix} \\ & =\frac{-3}{2}e^{\frac{t}{2}}-2e^{\frac{t}{2}}\\ & =\frac{-7}{2}e^{\frac{t}{2}} \end{align*}
Find the Wronskian of the given pair of functions \cos t,\sin t
solution
We are given y_{1}\left ( t\right ) =\cos t,y_{2}\left ( t\right ) =\sin t, hence by definition, the Wronskian is\begin{align*} W & =\begin{vmatrix} y_{1}\left ( t\right ) & y_{2}\left ( t\right ) \\ y_{1}^{\prime }\left ( t\right ) & y_{2}^{\prime }\left ( t\right ) \end{vmatrix} \\ & =\begin{vmatrix} \cos t & \sin t\\ -\sin t & \cos t \end{vmatrix} \\ & =\cos ^{2}t+\sin ^{2}t\\ & =1 \end{align*}
Find the Wronskian of the given pair of functions e^{-2t},te^{-2t}
solution
We are given y_{1}\left ( t\right ) =e^{-2t},y_{2}\left ( t\right ) =te^{-2t}, hence by definition, the Wronskian is\begin{align*} W & =\begin{vmatrix} y_{1}\left ( t\right ) & y_{2}\left ( t\right ) \\ y_{1}^{\prime }\left ( t\right ) & y_{2}^{\prime }\left ( t\right ) \end{vmatrix} \\ & =\begin{vmatrix} e^{-2t} & te^{-2t}\\ -2e^{-2t} & e^{-2t}-2te^{-2t}\end{vmatrix} \\ & =\left ( e^{-2t}\right ) \left ( e^{-2t}-2te^{-2t}\right ) +2e^{-2t}te^{-2t}\\ & =e^{-4t}-2te^{-4t}+2te^{-4t}\\ & =e^{-4t} \end{align*}
Find the Wronskian of the given pair of functions x,xe^{x}
solution
We are given y_{1}\left ( x\right ) =x,y_{2}\left ( x\right ) =xe^{x}, hence by definition, the Wronskian is\begin{align*} W & =\begin{vmatrix} y_{1}\left ( x\right ) & y_{2}\left ( x\right ) \\ y_{1}^{\prime }\left ( x\right ) & y_{2}^{\prime }\left ( x\right ) \end{vmatrix} \\ & =\begin{vmatrix} x & xe^{x}\\ 1 & e^{x}+xe^{x}\end{vmatrix} \\ & =\left ( x\right ) \left ( e^{x}+xe^{x}\right ) -xe^{x}\\ & =xe^{x}+x^{2}e^{x}-xe^{x}\\ & =x^{2}e^{x} \end{align*}
Find the Wronskian of the given pair of functions e^{t}\sin t,e^{t}\cos t
solution
We are given y_{1}\left ( t\right ) =e^{t}\sin t,y_{2}\left ( t\right ) =e^{t}\cos t, hence by definition, the Wronskian is\begin{align*} W & =\begin{vmatrix} y_{1}\left ( t\right ) & y_{2}\left ( t\right ) \\ y_{1}^{\prime }\left ( t\right ) & y_{2}^{\prime }\left ( t\right ) \end{vmatrix} \\ & =\begin{vmatrix} e^{t}\sin t & e^{t}\cos t\\ e^{t}\sin t+e^{t}\cos t & e^{t}\cos t-e^{t}\sin t \end{vmatrix} \\ & =\left ( e^{t}\sin t\right ) \left ( e^{t}\cos t-e^{t}\sin t\right ) -e^{t}\cos t\left ( e^{t}\sin t+e^{t}\cos t\right ) \\ & =e^{2t}\sin t\cos t-e^{2t}\sin ^{2}t-e^{2t}\cos t\sin t-e^{2t}\cos ^{2}t\\ & =-e^{2t}\sin ^{2}t-e^{2t}\cos ^{2}t\\ & =-2e^{2t}\left ( \sin ^{2}t+\cos ^{2}t\right ) \\ & =-2e^{2t} \end{align*}
Find the Wronskian of the given pair of functions \cos ^{2}\theta ,1+\cos 2\theta
solution
We are given y_{1}\left ( \theta \right ) =\cos ^{2}\theta ,y_{2}\left ( \theta \right ) =1+\cos 2\theta , hence by definition, the Wronskian is\begin{align*} W & =\begin{vmatrix} y_{1}\left ( \theta \right ) & y_{2}\left ( \theta \right ) \\ y_{1}^{\prime }\left ( \theta \right ) & y_{2}^{\prime }\left ( \theta \right ) \end{vmatrix} \\ & =\begin{vmatrix} \cos ^{2}\theta & 1+\cos 2\theta \\ -2\cos \theta \sin \theta & -2\sin 2\theta \end{vmatrix} \\ & =-2\cos ^{2}\theta \sin 2\theta -\left ( 1+\cos 2\theta \right ) \left ( -2\cos \theta \sin \theta \right ) \\ & =-2\cos ^{2}\theta \sin 2\theta -\left ( -2\cos \theta \sin \theta -2\cos \theta \sin \theta \cos 2\theta \right ) \\ & =-2\cos ^{2}\theta \sin 2\theta +2\cos \theta \sin \theta +2\cos \theta \sin \theta \cos 2\theta \end{align*}
Using \cos 2\theta =2\cos ^{2}\theta -1 And \sin 2\theta =2\sin \theta \cos \theta the above becomes\begin{align*} W & =-2\cos ^{2}\theta \left ( 2\sin \theta \cos \theta \right ) +2\cos \theta \sin \theta +2\cos \theta \sin \theta \left ( 2\cos ^{2}\theta -1\right ) \\ & =-4\cos ^{3}\theta \sin \theta +2\cos \theta \sin \theta +4\cos ^{3}\theta \sin \theta -2\cos \theta \sin \theta \\ & =-4\cos ^{3}\theta \sin \theta +4\cos ^{3}\theta \sin \theta \\ & =0 \end{align*}
We could also see that W=0 more directly, by noticing that y_{1}=\cos ^{2}\theta =1-\sin ^{2}\theta and since \sin ^{2}\theta =\frac{1}{2}-\frac{1}{2}\cos 2\theta then
\begin{align*} y_{1} & =\cos ^{2}\theta \\ & =1-\left ( \frac{1}{2}-\frac{1}{2}\cos 2\theta \right ) \\ & =\frac{1}{2}+\frac{1}{2}\cos 2\theta \\ & =\frac{1}{2}\left ( 1+\cos 2\theta \right ) \end{align*}
Therefore, y_{1}=\frac{1}{2}y_{2}. Hence y_{2} is just a scaled version of y_{1} and so these are two solutions are not linearly independent functions, (parallel to each others in vector space view) and so we expect that the Wronskian to be zero.