Question

Find the "general solution" of the equation \(y^{\prime}=\sqrt{y}\) by separation of variables. Find a particular solution satisfying \(y=0\) when \(x=0\). Show that the singular solution \(y=0\) cannot be obtained from the general solution. Sketch graphs of the "general solution" for several values of the arbitrary constant, and observe that each of them is tangent to the singular solution. Thus there are two solutions passing through any point on the \(x\) axis; in particular, there are two solutions satisfying \(x=y=0 .\) Problems 17 and 18 are physical problems leading to this differential equation.

Short answer

General solution is \( y = \frac{(x + C)^2}{4} \). The particular solution for \( y(0) = 0 \) is \( y = \frac{x^2}{4} \). Singular solution \( y = 0 \) cannot be obtained from the general solution.

Step-by-step solution

Write the Differential Equation

Start with the given differential equation: \[ y' = \sqrt{y} \] We will solve this using separation of variables.

Separate the Variables

Separate the variables such that all terms involving \(y\) are on one side and all terms involving \( x \) are on the other side. \[ \frac{dy}{\sqrt{y}} = dx \]

Integrate Both Sides

Integrate both sides of the equation: \[ \int \frac{1}{\sqrt{y}} dy = \int dx \] This results in: \[ 2 \sqrt{y} = x + C \] Where \( C \) is the integration constant.

Solve for y

Solve for \( y \) to find the general solution: \[ \sqrt{y} = \frac{x + C}{2} \] Square both sides: \[ y = \left( \frac{x + C}{2} \right)^2 \] Simplifying, we get: \[ y = \frac{(x + C)^2}{4} \]

Find Particular Solution for Given Conditions

To find a particular solution satisfying \( y(0) = 0 \), plug \( x = 0 \) and \( y = 0 \) into the general solution equation: \[ 0 = \frac{(0 + C)^2}{4} \] This simplifies to \[ C = 0 \]. Therefore, the particular solution is: \[ y = \frac{x^2}{4} \]

Confirm the Singular Solution

The singular solution is \( y = 0 \). Notice that for \( y = 0 \), the right-hand side of the original differential equation \( y' = \sqrt{y} \) is also 0. However, \( y = 0 \) cannot be derived from the general solution \( y = \frac{(x + C)^2}{4} \) because for any value of \( C \), the general solution does not satisfy \( y = 0 \) for any value of \( x \). Therefore, \( y = 0 \) is a unique solution not represented by the general solution.

Sketch the Graphs

Plot the general solutions \( y = \frac{(x + C)^2}{4} \) for several values of \( C \). Observe that each curve is parabolic in shape and is tangent to the singular solution \( y = 0 \) at exactly one point.

Key concepts

separation of variables

Separation of variables is a straightforward technique used to solve first-order differential equations where variables can be rearranged. Take the differential equation given in our problem: \( y' = \sqrt{y} \). The idea is to move all terms involving \( y \) to one side and all terms involving \( x \) to the other side:
\[ \frac{dy}{\sqrt{y}} = dx \] Now, each side of the equation can be integrated independently. This process allows us to solve the equation step by step.

It's like untangling two intertwined strings by moving them apart, making integration simpler.

particular solution

A particular solution is a specific solution derived from the general solution by applying initial conditions or boundary conditions. For example, given the condition \( y(0) = 0 \), substitute into the general solution to find a specific value for the constant:
\[ 0 = \frac{\left( 0 + C \right)^2}{4} \] Simplifying this gives us \( C = 0 \). Thus, the particular solution becomes:
\[ y = \frac{x^2}{4} \]
The particular solution uniquely fits the given initial conditions, providing a specific trajectory for the equation.

singular solution

A singular solution is a solution to a differential equation that is not part of the general solution family but still satisfies the equation. In our example, \( y = 0 \) is a singular solution.
It’s special because it cannot be obtained by any value of the integration constant \( C \) from the general solution \( y = \frac{(x + C)^2}{4} \). Singular solutions often reveal unique behaviors in differential equations, undetected by general solutions.
They might represent boundary behaviors or other special cases, providing deeper insights into the system's dynamics. Models can sometimes behave unexpectedly at singular points, so identifying these solutions is crucial.

integration constant

The integration constant \( C \) appears when we integrate an indefinite integral. It represents an entire family of solutions rather than just one specific solution.
In solving \( y' = \sqrt{y} \), we integrated both sides to get:
\[ 2 \sqrt{y} = x + C \]
This constant \( C \) signifies that there are infinitely many solutions based on different initial conditions. Each specific value of \( C \) corresponds to a different curve or trajectory in the \( y \)-\( x \)-plane.

graphical representation

Graphical representation involves plotting the solutions on a graph to visualize their behavior. For our general solution \( y = \frac{(x + C)^2}{4} \), each specific value of \( C \) will define a parabolic curve. Plotting these curves for several values of \( C \), you will notice:

• Each curve is parabolic.
• Each curve is tangent to the singular solution \( y = 0 \) at exactly one point.

This tangency is significant because it shows that while the curves approach the zero solution, they never fully align, emphasizing the uniqueness of the singular solution. Visualizing differential equations can often give better intuition of the solution behaviors.