3.2E: Exercises for Section 3.2

Exercise \(\PageIndex{1}\)

\(y''-3y'+2y=e^{3x}(1+x)\)

Exercise \(\PageIndex{2}\)

\(y''-6y'+5y=e^{-3x}(35-8x)\)

Answer

\(y_p = e^{-3x} \left( 1 - \frac{x}{4} \right).\)

Exercise \(\PageIndex{3}\)

\(y''-2y'-3y=e^x(-8+3x)\)

Exercise \(\PageIndex{4}\)

\(y''+2y'+y=e^{2x}(-7-15x+9x^2)\)

Answer

\(y_p = e^{2x} \left( 1 - 3x + x^2 \right).\)

Exercise \(\PageIndex{5}\)

\(y''+4y=e^{-x}(7-4x+5x^2)\)

Exercise \(\PageIndex{6}\)

\(y''-y'-2y=e^x(9+2x-4x^2)\)

Answer

\(y_p = e^x (-2 + x + 2x^2).\)

Exercise \(\PageIndex{7}\)

\(y''-4y'-5y=-6xe^{-x}\)

Exercise \(\PageIndex{8}\)

\(y''-3y'+2y=e^x(3-4x)\)

Answer

\(y_p = x e^x \left( 1 + 2x \right).\)

Exercise \(\PageIndex{9}\)

\(y''+y'-12y=e^{3x}(-6+7x)\)

Exercise \(\PageIndex{10}\)

\(2y''-3y'-2y=e^{2x}(-6+10x)\)

Exercise \(\PageIndex{11}\)

\(y''+2y'+y=e^{-x}(2+3x)\)

Exercise \(\PageIndex{12}\)

\(y''-2y'+y=e^x(1-6x)\)

Answer

\(y_p = x^2 e^x \left( \frac{1}{2} - x \right).\)

Exercise \(\PageIndex{13}\)

\(y''-4y'+4y=e^{2x}(1-3x+6x^2)\)

Exercise \(\PageIndex{14}\)

\(9y''+6y'+y=e^{-x/3}(2-4x+4x^2)\)

Exercise \(\PageIndex{15}\)

\(y''-3y'+2y=e^{3x}(1+x)\)

Exercise \(\PageIndex{16}\)

\(y''-6y'+8y=e^x(11-6x)\)

Answer

\(y = e^x (1 - 2x) + c_1 e^{2x} + c_2 e^{4x}\)

Exercise \(\PageIndex{17}\)

\(y''+6y'+9y=e^{2x}(3-5x)\)

Exercise \(\PageIndex{18}\)

\(y''+2y'-3y=-16xe^x\)

Answer

\(y = x e^x (1 - 2x) + c_1 e^x + c_2 e^{-3x}.\)

Exercise \(\PageIndex{19}\)

\(y''-2y'+y=e^x(2-12x)\)

Exercise \(\PageIndex{20}\)

\(y''-4y'-5y=9e^{2x}(1+x), \quad y(0)=0,\quad y'(0)=-10\)

Exercise \(\PageIndex{21}\)

\(y''+3y'-4y=e^{2x}(7+6x), \quad y(0)=2,\quad y'(0)=8\)

Exercise \(\PageIndex{22}\)

\(y''+4y'+3y=-e^{-x}(2+8x), \quad y(0)=1,\quad y'(0)=2\)

Answer

\(y = e^{-x} (2 + x - 2x^2) - e^{-3x}.\)

Exercise \(\PageIndex{23}\)

\(y''-3y'-10y=7e^{-2x}, \quad y(0)=1,\quad y'(0)=-17\)

Exercise \(\PageIndex{24}\)

\(y''+y'+y=xe^x+e^{-x}(1+2x)\)

Exercise \(\PageIndex{25}\)

\(y''-7y'+12y=-e^x(17-42x)-e^{3x}\)

Exercise \(\PageIndex{26}\)

\(y''-8y'+16y=6xe^{4x}+2+16x+16x^2\)

Exercise \(\PageIndex{27}\)

\(y''-3y'+2y=-e^{2x}(3+4x)-e^x\)

Exercise \(\PageIndex{28}\)

\(y''-2y'+2y=e^x(1+x)+e^{-x}(2-8x+5x^2)\)

Answer

\(y_p = y_{p_1} + y_{p_2} = e^x (1 + x) + x^2 e^{-x}.\)

Exercise \(\PageIndex{29}\)

\(y''+y=e^{-x}(2-4x+2x^2)+e^{3x}(8-12x-10x^2)\)

Exercise \(\PageIndex{30}\)

1. Prove that \(y\) is a solution of the constant coefficient equation \[ay''+by'+cy=e^{\alpha x}G(x) \tag{A} \] if and only if \(y=ue^{\alpha x}\), where \(u\) satisfies \[au''+p'(\alpha)u'+p(\alpha)u=G(x) \tag{B} \] and \(p(r)=ar^2+br+c\) is the characteristic polynomial of the complementary equation \[ay''+by'+cy=0.\nonumber \] For the rest of this exercise, let \(G\) be a polynomial. Give the requested proofs for the case where \[G(x)=g_0+g_1x+g_2x^2+g_3x^3.\nonumber \]
2. Prove that if \(e^{\alpha x}\) isn’t a solution of the complementary equation then (B) has a particular solution of the form \(u_p=A(x)\), where \(A\) is a polynomial of the same degree as \(G\), as in Example 3.2.4. Conclude that (A) has a particular solution of the form \(y_p=e^{\alpha x}A(x)\).
3. Show that if \(e^{\alpha x}\) is a solution of the complementary equation and \(xe^{\alpha x}\) isn’t , then (B) has a particular solution of the form \(u_p=xA(x)\), where \(A\) is a polynomial of the same degree as \(G\), as in Example 3.2.5. Conclude that (A) has a particular solution of the form \(y_p=xe^{\alpha x}A(x)\).
4. Show that if \(e^{\alpha x}\) and \(xe^{\alpha x}\) are both solutions of the complementary equation then (B) has a particular solution of the form \(u_p=x^2A(x)\), where \(A\) is a polynomial of the same degree as \(G\), and \(x^2A(x)\) can be obtained by integrating \(G/a\) twice, taking the constants of integration to be zero, as in Example 3.2.6. Conclude that (A) has a particular solution of the form \(y_p=x^2e^{\alpha x}A(x)\).

Exercise \(\PageIndex{31}\)

Compare with Example 3.2.1:

\[y''-7y'+12y=4e^{2x};\quad y_p=Ae^{2x}\nonumber \]

Exercise \(\PageIndex{32}\)

Compare with Example 3.2.2:

\[y''-7y'+12y=5e^{4x};\quad y_p=Axe^{4x}\nonumber \]

Exercise \(\PageIndex{33}\)

Compare with Example 3.2.3:

\[y''-8y'+16y=2e^{4x};\quad y_p=Ax^2e^{4x}\nonumber \]

Exercise \(\PageIndex{34}\)

Compare with Example 3.2.4:

\[y''-3y'+2y=e^{3x}(-1+2x+x^2),\quad y_p=e^{3x}(A+Bx+Cx^2)\nonumber \]

Answer

\(y_p = -\frac{e^{3x}}{4} \left( 1 + 2x - 2x^2 \right).\)

Exercise \(\PageIndex{35}\)

Compare with Example 3.2.5:

\[y''-4y'+3y=e^{3x}(6+8x+12x^2),\quad y_p=e^{3x}(Ax+Bx^2+Cx^3)\nonumber \]

Exercise \(\PageIndex{36}\)

Compare with Example 3.2.6:

\[4y''+4y'+y=e^{-x/2}(-8+48x+144x^2),\quad y_p=e^{-x/2}(Ax^2+Bx^3+Cx^4)\nonumber \]

Exercise \(\PageIndex{37}\)

Write \(y=ue^{\alpha x}\) to find the general solution.

1. \(y''+2y'+y={e^{-x}\over\sqrt x}\)
2. \(y''+6y'+9y=e^{-3x}\ln x\)
3. \(y''-4y'+4y={e^{2x}\over1+x}\)
4. \(4y''+4y'+y={4e^{-x/2}\left({1\over x}+x\right)}\)

Exercise \(\PageIndex{38}\)

Suppose \(\alpha\ne0\) and \(k\) is a positive integer. In most calculus books integrals like \(\int x^k e^{\alpha x}\,dx\) are evaluated by integrating by parts \(k\) times. This exercise presents another method. Let

\[y=\int e^{\alpha x}P(x)\,dx\nonumber \]

with

\[P(x)=p_0+p_1x+\cdots+p_kx^k\nonumber \]

(where \(p_k \neq 0\)).

1. Show that \(y=e^{\alpha x}u\), where \[u'+\alpha u=P(x). \tag{A} \]
2. Show that (A) has a particular solution of the form \[u_p=A_0+A_1x+\cdots+A_kx^k,\nonumber \] where \(A_k\), \(A_{k-1}\), …, \(A_0\) can be computed successively by equating coefficients of \(x^k,x^{k-1}, \dots,1\) on both sides of the equation \[u_p'+\alpha u_p=P(x).\nonumber \]
3. Conclude that \[\int e^{\alpha x}P(x)\,dx=\left(A_0+A_1x+\cdots+A_kx^k\right)e^{\alpha x} +c,\nonumber \] where \(c\) is a constant of integration.

Exercise \(\PageIndex{39}\)

Use the method suggested in Exercise 3.2E.38 to evaluate the integrals.

1. \(\int e^{x}(4+x)dx\)
2. \(\int e^{-x}(-1+x^{2})dx\)
3. \(\int x^{3}e^{-2x}dx\)
4. \(\int e^{x}(1+x)^{2}dx\)
5. \(\int e^{3x}(-14+30x+27x^{2})dx\)
6. \(\int e^{-x}(1+6x^{2}-14x^{3}+3x^{4})dx\)

Exercise \(\PageIndex{40}\)

Use the method suggested in Exercise 3.2E.38 to evaluate \(\int x^ke^{\alpha x}\,dx\), where \(k\) is an arbitrary positive integer and \(\alpha\ne0\).