Differential equations

Unit 5 Differential equations

Key unit competence: Apply ordinary differential equations of first and second order in solving real life problems.

Key ideas

• A differential equation links a function and its derivatives. The order is the highest derivative; the degree is the power of that highest derivative. A DE is formed by eliminating arbitrary constants.
• Separable $\frac{dy}{dx}=g(x)h(y)$: write $\frac{dy}{h(y)}=g(x)dx$ and integrate both sides.
• Homogeneous first order $y^{\prime }=F\left(\frac{y}{x}\right)$: put $y=vx$, so $y^{\prime }=v+x{v}^{\prime }$, which becomes separable.
• Linear first order $y^{\prime }+P(x)y=Q(x)$: integrating factor $\mu =e^{\int Pdx}$, then $(\mu y{)}^{\prime }=\mu Q$.
• Second order $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$: characteristic equation $a{r}^2+br+c=0$. Two real roots: $y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$. Double root: $y=(C_1+C_{2}x)e^{rx}$. Complex roots $\alpha \pm i\beta$: $y=e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x)$.
• Non-homogeneous: $y=y_h+y_p$. If the right side is $P(x)e^{\alpha x}$, try $y_p=x^{k}Q(x)e^{\alpha x}$ ($Q$ same degree as $P$, $k$ = number of times $\alpha$ is a root). If it is $e^{\alpha x}(P\cos \beta x+Q\sin \beta x)$, try $x^k{e}^{\alpha x}(U\cos \beta x+V\sin \beta x)$, $k=1$ if $\alpha +i\beta$ is a root.
• Applications: growth and decay $y^{\prime }=ky$; Newton's cooling $T^{\prime }=-k(T-T_s)$; electric circuits $L{i}^{\prime }+Ri=E$; springs $m{y}^{\prime \prime }+ky=0$.

Formulas to know

Common misconceptions

• Forgetting the arbitrary constant, then being unable to use the initial condition.
• With a double root, writing $C_1{e}^{rx}+C_2{e}^{rx}$ (only one constant). Use $(C_1+C_{2}x)e^{rx}$.
• Trying $y_p=A{e}^{\alpha x}$ when $\alpha$ is already a root of the characteristic equation: multiply by $x$.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Form the differential equation whose general solution is $y=A{e}^{2x}+B{e}^{-x}$.

   Show answer

   The roots 2 and $-1$ come from $r^2-r-2=0$, so $y^{\prime \prime }-y^{\prime }-2y=0$.

   Common mistake: Eliminating only one constant.

2. In an RL circuit, $2\frac{di}{dt}+4i=12$ and $i(0)=0$. Find $i(t)$ and the steady current.

   Show answer

   $i^{\prime }+2i=6$: $i=3+C{e}^{-2t}$, $C=-3$: $i=3(1-e^{-2t})$. As $t\to \infty$, $i\to 3$ A.

   Common mistake: Ignoring the initial condition.

3. Solve $(1+x^2)y^{\prime }=xy$ with $y(0)=1$.

   Show answer

   $\frac{dy}{y}=\frac{xdx}{1+x^2}$, $\ln y=\frac{1}{2}\ln (1+x^2)$, so $y=\sqrt{1+x^2}$.

   Common mistake: Forgetting to use the initial condition.

4. Solve $y^{\prime \prime }+y=e^x$.

   Show answer

   $y_h=C_1\cos x+C_2\sin x$; $y_p=A{e}^x$ gives $2A=1$: $y=C_1\cos x+C_2\sin x+\frac{1}{2}{e}^x$.

   Common mistake: Writing $y_h=C_1{e}^x+C_2{e}^{-x}$ (that is for $y^{\prime \prime }-y=0$).

5. Solve $y^{\prime }=\frac{x+y}{x}$ for $x>0$.

   Show answer

   $y=vx$: $v+x{v}^{\prime }=1+v$, so $v^{\prime }=\frac{1}{x}$, $v=\ln x+C$, $y=x\ln x+Cx$.

   Common mistake: Writing $y^{\prime }=v^{\prime }$ instead of $v+x{v}^{\prime }$.

6. Solve $y^{\prime \prime }-3y^{\prime }+2y=4x$.

   Show answer

   $y_h=C_1{e}^x+C_2{e}^{2x}$. Try $y_p=ax+b$: $-3a+2ax+2b=4x$, so $a=2$, $b=3$. $y=C_1{e}^x+C_2{e}^{2x}+2x+3$.

   Common mistake: Trying $y_p=ax$ only (missing the constant term).

7. Solve $y^{\prime }-2y=4$ with $y(0)=1$.

   Show answer

   $y=-2+C{e}^{2x}$; $y(0)=1$ gives $C=3$: $y=3e^{2x}-2$.

   Common mistake: Forgetting the constant solution part $-2$.

8. Solve $y^{\prime \prime }+6y^{\prime }+9y=0$ with $y(0)=1$, $y^{\prime }(0)=0$.

   Show answer

   Double root $-3$: $y=(C_1+C_{2}x)e^{-3x}$. $C_1=1$, $y^{\prime }(0)=C_2-3C_1=0$, $C_2=3$: $y=(1+3x)e^{-3x}$.

   Common mistake: Differentiating without the product rule.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find a particular solution of $y^{\prime \prime }+y=x^2$.	a. $y=2-2e^{-2x}$
   2. Find the general solution of $y^{\prime \prime }+9y=0$.	b. $y_p=x^2-2$
   3. Find the general solution of $y^{\prime \prime }-5y^{\prime }+6y=0$.	c. $y=A\cos 3x+B\sin 3x$
   4. Solve $y^{\prime }+2y=4$ with $y(0)=0$.	d. $y=A{e}^{2x}+B{e}^{3x}$

   Show answer

   1 → b ($y_p=x^2-2$) · 2 → c ($y=A\cos 3x+B\sin 3x$) · 3 → d ($y=A{e}^{2x}+B{e}^{3x}$) · 4 → a ($y=2-2e^{-2x}$)

   • Step 1. Try $y_p=a{x}^2+bx+c$: $y_p^{\prime \prime }=2a$.
     Step 2. $a{x}^2+bx+(c+2a)=x^2\Rightarrow a=1$, $b=0$, $c=-2$.
   • Step 1. $r^2+9=0\Rightarrow r=\pm 3i$.
     Step 2. Complex roots $\pm \beta i$ give $A\cos \beta x+B\sin \beta x$ with $\beta =3$.
   • Step 1. Characteristic equation: $r^2-5r+6=0$.
     Step 2. $(r-2)(r-3)=0$, so $r=2$, $r=3$.
   • Step 1. $(y{e}^{2x}{)}^{\prime }=4e^{2x}\Rightarrow y{e}^{2x}=2e^{2x}+C\Rightarrow y=2+C{e}^{-2x}$.
     Step 2. $y(0)=0$: $C=-2$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $(C_1+C_{2}x)e^x+x{e}^x$  ·  $(C_1+C_{2}x)e^x+\frac{1}{2}{x}^2{e}^x$  ·  $y=x^2+Cx$  ·  $\frac{x}{4}\sin 2x$
   1. The general solution of $y^{\prime \prime }-2y^{\prime }+y=e^x$ is .
   2. A particular solution of $y^{\prime \prime }+4y=\cos 2x$ is .
   3. For $x>0$, the general solution of $x{y}^{\prime }-y=x^2$ is .
   Show answer

   1: $(C_1+C_{2}x)e^x+\frac{1}{2}{x}^2{e}^x$ · 2: $\frac{x}{4}\sin 2x$ · 3: $y=x^2+Cx$

   • Step 1. $r^2-2r+1=(r-1{)}^2$: complementary $(C_1+C_{2}x)e^x$.
     Step 2. 1 is a double root, so try $y_p=A{x}^2{e}^x$: substitution gives $2A{e}^x=e^x$, $A=\frac{1}{2}$.
   • Step 1. $\pm 2i$ are roots of $r^2+4=0$ (resonance), so try $y_p=x(A\cos 2x+B\sin 2x)$.
     Step 2. Substituting gives $-4A\sin 2x+4B\cos 2x=\cos 2x$: $A=0$, $B=\frac{1}{4}$.
   • Step 1. Divide by $x$: $y^{\prime }-\frac{y}{x}=x$; integrating factor $\frac{1}{x}$.
     Step 2. ${\left(\frac{y}{x}\right)}^{\prime }=1\Rightarrow \frac{y}{x}=x+C\Rightarrow y=x^2+Cx$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find an integrating factor of $y^{\prime }+\frac{2}{x}y=x$ ($x>0$).	a. $y=x^3+2x+1$
   2. Find the general solution of $y^{\prime }-y=2x$.	b. $y=Cx$
   3. Solve $y^{\prime \prime }=6x$ with $y(0)=1$ and $y^{\prime }(0)=2$.	c. $y=C{e}^x-2x-2$
   4. Find the general solution of $\frac{dy}{dx}=\frac{y}{x}$.	d. $x^2$

   Show answer

   1 → d ($x^2$) · 2 → c ($y=C{e}^x-2x-2$) · 3 → a ($y=x^3+2x+1$) · 4 → b ($y=Cx$)

   • Step 1. $\mu =e^{\int \frac{2}{x}dx}=e^{2\ln x}$.
     Step 2. $=x^2$.
   • Step 1. Homogeneous part: $C{e}^x$. Try $y_p=ax+b$: $a-(ax+b)=2x$.
     Step 2. $-a=2\Rightarrow a=-2$; $a-b=0\Rightarrow b=-2$.
   • Step 1. $y^{\prime }=3x^2+C_1$; $y^{\prime }(0)=2\Rightarrow C_1=2$.
     Step 2. $y=x^3+2x+C_2$; $y(0)=1\Rightarrow C_2=1$.
   • Step 1. Separate: $\frac{dy}{y}=\frac{dx}{x}$.
     Step 2. $\ln |y|=\ln |x|+k\Rightarrow y=Cx$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   22.4 min  ·  $-2x$  ·  30 min  ·  $y=2e^x+e^{-2x}$
   1. A particular solution of $y^{\prime \prime }-y=2x$ is .
   2. The solution of $y^{\prime \prime }+y^{\prime }-2y=0$ with $y(0)=3$, $y^{\prime }(0)=0$ is .
   3. Tea at 90°C cools to 60°C in 10 minutes in a 20°C room. It reaches 40°C after about .
   Show answer

   1: $-2x$ · 2: $y=2e^x+e^{-2x}$ · 3: 22.4 min

   • Step 1. Try $y_p=ax+b$: $y_p^{\prime \prime }=0$, so $-(ax+b)=2x$.
     Step 2. $a=-2$, $b=0$: $y_p=-2x$.
   • Step 1. $r^2+r-2=(r-1)(r+2)$: $y=A{e}^x+B{e}^{-2x}$.
     Step 2. $A+B=3$ and $A-2B=0$: $B=1$, $A=2$.
   • Step 1. $T-20=70e^{-kt}$; at $t=10$: $40=70e^{-10k}$, so $e^{-10k}=\frac{4}{7}$.
     Step 2. $70e^{-kt}=20\Rightarrow e^{-kt}=\frac{2}{7}$: $t=10\times \frac{\ln 3.5}{\ln 1.75}\approx 22.4$ min.

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