# In Problem 1 we saw that cos x and e x were solutions of the nonlinear equation

## Question:

In Problem 1 we saw that cos x and e_{x} were solutions of the nonlinear equation (y'')^{2} - y^{2} = 0. Verify that sin x and e^{-x} are also solutions. Without attempting to solve the differential equation, discuss how these explicit solutions can be found by using knowledge about linear equations. Without attempting to verify, discuss why the linear combinations y = c_{1}e^{x} + c_{2}e^{-x} + c_{3} cos x + c_{4 }sin x and y = c_{2}e^{-x} + c_{4} sin x are not, in general, solutions, but the two special linear combinations y = c_{1}e^{x} + c_{2}e^{-x }and y = c_{3} cos x + c_{4} sin x must satisfy the differential equation.

Fantastic news! We've Found the answer you've been seeking!

## Step by Step Answer:

**Related Book For**

## A First Course in Differential Equations with Modeling Applications

**ISBN:** 978-1111827052

10th edition

**Authors:** Dennis G. Zill

**Question Posted:**