Solve the differential equation (dy/dx) 2 = 4y to verify the general solution curves and singular solution
Question:
Solve the differential equation (dy/dx)2 = 4y to verify the general solution curves and singular solution curve that are illustrated in Fig. 1.4.5. Then determine the points (a, b) in the plane for which the initial value problem (y')2 = 4y, y(a) = b has (a) no solution,
(b) Infinitely many solutions that are defined for all x,
(c) On some neighborhood of the point x = a, only finitely many solutions.
Fig. 1.4.5
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Step by Step Answer:
To solve the differential equation dydx2 4y and verify the general and singular solution curves we can follow these steps Step 1 Rewrite the differential equation in a standard form dydx2 4y 0 Step 2 Factorize the equation dydx 2ydydx 2y 0 Step 3 Set each factor equal to zero and solve for dydx dydx 2y 0 dydx 2y Equation 1 dydx 2y 0 dydx 2y Equation 2 Step 4 Solve the first equation Equation 1 dyy 2dx Integrate both sides 2y 2x C1 where C1 is the constant of integration Simplify y x C12 Square both sides y x C122 Equation 3 Step 5 Solve the second equation Equation 2 dyy 2dx Integrate both sides 2y 2x C2 where C2 is the constant of integration Simplify y x C22 Square both sides y x C222 Equation 4 Equations 3 and 4 represent the general solution curves for the given differential equation To determine the points a b in the plane for which the initial value problem y2 4y ya b has a No solution This occurs when the square root of y becomes undefined or negative Therefore for no solution y must be negative or imaginary In the given differential equation y is always nonnegative so there are no points a b for which this condition is satisfied b Infinitely many solutions that are defined for all x This occurs when the constants C1 and C2 are chosen such that the solutions cover all possible values of y In Equation 3 if C1 0 then y x2 which represents a parabolic curve that covers all nonnegative values of y Therefore for infinitely many solutions defined for all x any point a b where b 0 is satisfied c Only finitely many solutions on some neighborhood of the point x a This occurs when the solutions are limited to a specific range or interval In Equation 4 if C2 0 then y x2 which represents a parabolic curve similar to Equation 3 but only for nonnegative values of y Therefore for finitely many solutions on some neighborhood of the point x a any point a b where 0 b is satisfied By analyzing the general solution curves and considering the conditions for different ...View the full answer
Related Book For 
Differential Equations And Linear Algebra
ISBN: 9780134497181
4th Edition
Authors: C. Edwards, David Penney, David Calvis
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