Consider the differential equation \(25 \frac{dy}{dx} - 36x = 0\), which has a one-parameter family of implicit solutions \(36x^2 - 25y^2 = C\) for every constant \(C\). Complete parts (a) through (c) below.### (a) Does the Existence and Uniqueness of Solution Theorem imply the existence of a unique solution to the differential equation that satisfies \(y(0) = 0\)? Select the correct choice below and, if necessary, fill in the answer box within your choice.- ⭕ A. The theorem implies the existence of a unique solution for all values of \(x_0\).- ⭕ B. The theorem implies the existence of a unique solution only for values of \(x_0\) in the interval \(\_\_\_\) (Type your answer in interval notation.)- ⭕ C. The theorem does not imply the existence of a unique solution for any values of \(x_0\).### (b) Show that when \(x_0 \neq 0\), the differential equation can’t possibly have a solution in a neighborhood of \(x = x_0\) that satisfies \(y(x_0) = 0\).Substituting the initial condition into the differential equation gives \[ 25 \frac{dy}{dx} - 36x ( ) = 0 \]This equation can be simplified to \[\_\_\_\_\_\_\_\_\] which \[\_\_\_\_\_\_\_\_\] the given condition that \[\_\_\_\_\_\_\_\_\].### (c) Show that there are two distinct solutions to the differential equation satisfying \(y(0) = 0\). Refer to the graph showing several members of the solution family \(36x^2 - 25y^2 = C\).The initial condition \(y(0) = 0\) leads to the solution(s) with \(C = \_\_\_\_\), which gives two explicit solutions \(y = \_\_\_\_\_\_\_\_\), that satisfy both the differential equation and the initial condition. (Use a comma to separate answers as needed.)_**Graph Explanation:**_The graph to the right displays several members of the solution family \(36x^2 - 25y^2 = C\), with different values of

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dy Consider the differential equation 25y36x=0, which has a one-parameter family of implicit solutions 36x² - 25y² = C for every constant C. Complete parts (a) through (c) below. (a) Does the Existence and Uniqueness Solution Theorem imply the existence of a unique solution to the differential equation that satisfies y(x)=0? Select the correct choice below and, if necessary, fill in the answer box within your choice. OA. The theorem implies the existence of a unique solution for all values of X- OB. The theorem implies the existence of a unique solution only for values of xo in the interval (Type your answer in interval notation.) O c. The theorem does not imply the existence of a unique solution for any values of X (b) Show that when x * 0, the differential equation can't possibly have a solution in a neighborhood of x= x that satisfies y (xo) = 0. dy Substituting the initial condition into the differential equation gives 25 (-36= 0. This equation can be simplified to dx ▼which the given condition that (c) Show that there are two distinct solutions to the differential equation satisfying y(0) = 0. Refer to the graph showing several members of the solution family 36x² - 25y² = C. The initial condition y(0) = 0 leads to the solution (s) with C= which gives two explicit solutions y=.that satisfy both the differential equation and the initial condition. (Use a comma to separate answers as needed.) ▼ C=-100 C=36 C=-25 C=144 X * -2/ C=0 C=0

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter11: Differential Equations

Section11.1: Solutions Of Elementary And Separable Differential Equations

Problem 59E: According to the solution in Exercise 58 of the differential equation for Newtons law of cooling,...