$$\begin{array}{l}{y^{\prime \prime}(\theta)+y(\theta)^{3}=\sin \theta} \\ {y(0)=0, \quad y^{\prime}(0)=0}\end{array}$$

Answer

$$
y(\theta)=\frac{\theta^{3}}{6}-\frac{\theta^{5}}{120}+\frac{\theta^{7}}{5040}
$$

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Calculus 1 / AB

Fundamentals of Differential Equations

Chapter 8

Series Solutions of Differential Equations

Section 1

Introduction: The Taylor Polynomial Approximation

Discussion

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Problem 16

Video Transcript

everyone today we're looking at the differential equation. Why? Prime Prime was t Y prime minus y is equal to zero. We're trying to solve this equation with the initial conditions. Why have 00? Why problem is era is three. Now, if we take the last last transform of both sides of this equation, we get the lap. Last transform of white prime prime is s squared capital y. Let's say that the lacked last transfer then squared y minus three is the lack last transform? What print prime? The lack last ransom of tea times Wife prime is the last last transform of why prime, which is why differentiated but must transfer of lower case wise capital. Why? On the left hand, Trump Transformer zero is zero. Um, Now, if you rearrange this equation, we find that X squared minus two times why minus f times why prime is equal to three. This is the first order differential equation in capital. Why? So we can solve it using standard techniques in first order differential equations. In particular, we can use the method of integrating factors to solve this equation in capital. Why in particular we find that one solution is why is equal to three over s squared, and now we take the inverse lab last transform. We find that why is equal to three teeth?

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