Adjust the initial condition in the interactive to help you answer the following:
(a) Find the particular solution that passes through \((0,5)\).
Use the figure to determine the particular solution of \(y = ce^{x^2} + 3\) that passes through the point \((0,5)\text{?}\)
- \(\quad\ds y = -2e^{x^2}+3\)
- Incorrect. The value of \(c\) must make the solution pass through \((0, 5)\text{.}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
- \(\quad\ds y = 0.5e^{x^2}+3\)
- Incorrect. Remember that at \(x = 0\text{,}\) the exponential term \(e^{x^2}\) equals 1, so \(y(0) = c + 3\text{.}\) What value of \(c\) gives \(y(0) = 5\text{?}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
- \(\quad\ds y = 2e^{x^2}+3\)
- Correct! The value \(c = 2\) ensures that \(y(0) = 2 + 3 = 5\text{,}\) so this solution passes through \((0, 5)\text{.}\)
- \(\quad\ds y = 5e^{x^2}+3\)
- Incorrect. The general solution would pass through \((0, 4)\) if \(c = 1\text{.}\) Hover over the curve in the figure that passes through \((0,5)\text{.}\)
(b) Find \(c\) for the particular solution that passes through \((1,1)\).
Adjust the initial condition to determine the value of \(c\) that corresponds to the particular solution that passes through the point \((1,1)\text{?}\)
- \(\quad c = -2\)
- Incorrect. Hover over the curve in the figure that passes through \((1,1)\) and look at the coefficient on \(e^{x^2}\text{.}\)
- \(\quad c = 0.5\)
- Incorrect. Hover over the curve in the figure that passes through \((1,1)\) and look at the coefficient on \(e^{x^2}\text{.}\)
- \(\quad c = 2\)
- Incorrect. Hover over the curve in the figure that passes through \((1,1)\) and look at the coefficient on \(e^{x^2}\text{.}\)
- \(\quad c = -1\)
- Correct! Hovering over the curve passing through \((1,1)\) shows the particular solution \(\ds y = -e^{x^2}+3\text{,}\) so \(c=1\text{.}\)
(c) Does the solution with \(c=-1\) pass through the origin?
Does the particular solution corresponding to \(c = -1\) pass through the origin, \((0, 0)\text{?}\)
You can answer this with or without adjusting the figure.
- Yes
- Incorrect. Try recalculating \(y(0)\) when \(c = -1\text{.}\) Which point does it pass through on the \(y\)-axis?
- No
- Correct. If \(c = -1\text{,}\) then \(y(0) = -1 + 3 = 2\text{,}\) which passes through \((0,2)\text{.}\)
(d) A solution that satisfies \(y(1) = 3\) also satisfies which condition?
A solution that satisfies \(y(1) = 3\) also satisfies which condition?
- \(\quad y(5)=3\)
- Correct! The graph of the solution that satisfies \(y(1) = 3\) is a horizontal line at \(y=3\) so it satisfies \(y(5) = 3\text{.}\)
- \(\quad y(1)=0\)
- Incorrect. The solution does not satisfy \(y(1)=0\text{.}\)
- \(\quad y(0)=0\)
- Incorrect. The solution does not satisfy \(y(0)=0\text{.}\)
- \(\quad y(1)=-3\)
- Incorrect. The solution does not satisfy \(y(1)=-3\text{.}\)
(e) Role of Initial Conditions.
What role do initial conditions play in solving differential equations?
- They determine the general form of the solution.
- Incorrect. Initial conditions are not used to find the general solution.
- They used to determine the constants in the general solution.
- Correct! Initial conditions are used to find specific values for constants in the general solution.
- They are used to find the particular solution.
- Correct! Initial conditions are used to find the specific solution that applies to a particular scenario.
- They are not needed if the general solution is already known.
- Incorrect. If provided, initial conditions are always needed to get the particular solution from the general solution.
