For the given differential equation xy'' - 2(x + 1)y' + (x + 2)y = 0, (a) Show that y = e^x is a solution. (b) Find a second solution and write down the general solution.
Added by Christopher T.
Step 1
First derivative: $y' = \frac{d}{dx}(e^x) = e^x$ Second derivative: $y'' = \frac{d^2}{dx^2}(e^x) = e^x$ Now, plug these derivatives into the given differential equation: $xy'' - 2(x + 1)y' + (x + 2)y = x(e^x) - 2(x + 1)(e^x) + (x + 2)(e^x)$ Simplify the Show more…
Show all steps
Close
Your feedback will help us improve your experience
Madhur L and 82 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Show that $y=x$ is a solution of the differential equation $x^{2} y^{\prime \prime}-\left(2 x+x^{2}\right) y^{\prime}+(2+x) y=0,$ and find the general solution of this equation.
Ordinary Differential Equations
Differential Equations of Second Order
Find the particular solution to the differential equation $y^{\prime}=2 x y$ that passes through $\left(0, \frac{1}{2}\right), \quad$ given that $y=C e^{x^{2}}$ is a general solution.
Introduction to Differential Equations
Basics of Differential Equations
(a) Show that each equation is an exact differential equation. (b) Find the general solution. $\frac{1}{y} d x-\frac{x}{y^{2}} d y=0$
Differential Equations
Exact Differential Equations
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD
