MHB Verifying a Solution to an ODE: Differential Equations HW Help!

[MarkFL]

Here is the question:

Differential Equations HW HELP!?

13. y'' + y = sec(t), 0< t < π/2; y = (cos t) ln cos t + t sin t

In each of Problems 7 through 14, verify that each given function is a solution of the differential equation.

I'm currently trying to review derivatives, but trig derivatives are super hard. Help?

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

Hello Clara I,

If the given solution is indeed a solution to the ODE, then we should find by adding the second derivative of the solution to itself, gives $\sec(t)$.

In order to differentiate the given solution with respect to $t$, we will need the following rules:

The Product Rule:

$$\frac{d}{du}\left(f(u)g(u) \right)=f(u)\frac{d}{du}(g(u))+\frac{d}{du}(f(u))g(u)$$

The Logarithmic Rule:

$$\frac{d}{du}(\ln(u))=\frac{1}{u}$$

The Chain Rule:

$$\frac{d}{du}\left(f(g(u)) \right)=\frac{df}{dg}\cdot\frac{dg}{du}$$

Trigonometric Rules:

$$\frac{d}{du}\left(\sin(u) \right)=\cos(u)$$

$$\frac{d}{du}\left(\cos(u) \right)=-\sin(u)$$

Using these rules, we may now compute:

$$y=\cos(t)\ln\left(\cos(t) \right)+t\sin(t)$$

$$\frac{dy}{dt}=\cos(t)\cdot\frac{-\sin(t)}{\cos(t)}-\sin(t)\ln\left(\cos(t) \right)+t\cos(t)+\sin(t)=t\cos(t)-\sin(t)\ln\left(\cos(t) \right)$$

$$\frac{d^2y}{dt^2}=-t\sin(t)+\cos(t)+\frac{\sin^2(t)}{\cos(t)}-\cos(t)\ln\left(\cos(t) \right)=-t\sin(t)+\frac{\cos^2(t)+\sin^2(t)}{\cos(t)}-\cos(t)\ln\left(\cos(t) \right)=$$

$$-t\sin(t)+\sec(t)-\cos(t)\ln\left(\cos(t) \right)$$

And so we find:

$$\frac{d^2y}{dt^2}+y=-t\sin(t)+\sec(t)-\cos(t)\ln\left(\cos(t) \right)+\cos(t)\ln\left(\cos(t) \right)+t\sin(t)=\sec(t)$$

Thus, we have shown the given solution does in fact satisfy the given ODE.