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Chapter 20: Problem 2

By integrating twice find the general solution of \(y^{\prime \prime}=12 x^{2}\)

Short Answer

Expert verified
Answer: The general solution of the given second-order differential equation is \(y(x) = x^4 + C_1x + C_2\), where \(C_1\) and \(C_2\) are constants of integration.

Step by step solution

01

Find \(y'\) by integrating the given equation with respect to \(x\)

To find \(y'\), integrate the given equation with respect to \(x\): $$ y' = \int 12x^2 \, dx $$ You can use polynomial integration rules to solve this integral: $$ y' = 4x^3 + C_1 $$ Here, \(C_1\) is the constant of integration.
02

Integrate \(y'\) once more to find \(y\)

Now, we will find the function \(y(x)\) by integrating \(y'\) with respect to \(x\): $$ y(x) = \int (4x^3 + C_1) \, dx $$ Again, use polynomial integration rules to solve this integral: $$ y(x) = x^4 + C_1x + C_2 $$ Here, \(C_2\) is another constant of integration.
03

Write the general solution

The general solution of the given second-order differential equation is: $$ y(x) = x^4 + C_1x + C_2 $$ Where \(C_1\) and \(C_2\) are constants of integration.

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Most popular questions from this chapter

Use Euler's method to find a numerical solution of \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-2 y, y(0)=1\), for \(0 \leq x \leq 0.5 .\) First take \(h=0.1\), then \(h=0.05\), and compare your answers at \(x=0.5\) with the exact solution obtained by separating the variables. Work throughout to six decimal places.

Find a second-order differential equation that is satisfied by $$ y=A \cosh 2 x+B \sinh 2 x $$

Identify the dependent and independent variables of the following differential equations. Give the order of the equations and state which are linear. (a) \(\frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\) (b) \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)+3 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\) (c) \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+5 \frac{\mathrm{d} x}{\mathrm{~d} t}=\sin x\)

Integrate the equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{2}\) subject to the condition \(y(1)=4\) in order to find the particular solution.

Integrate twice the differential equation $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{w}{2}\left(l x-x^{2}\right) $$ where \(w\) and \(l\) are constants, to find a general solution for \(y\).
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