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   <text:p text:style-name="Standard">DAE of simple pendulum</text:p>
   <text:p text:style-name="Standard">Define state variables:</text:p>
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z_2 = y 
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z_3 = dot x 
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&quot;&quot;
z_4 = dot y 
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&quot;&quot;
z_5 = %lambda 

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   <text:p text:style-name="Standard">The DAE in semi-explicit form:</text:p>
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newline
&quot;&quot;
{dot z}_2 = z_4 
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&quot;&quot;
{dot z}_3 = -z_1 z_5 
newline
&quot;&quot;
{dot z}_4 = -z_2 z_5 - g 
newline
&quot;&quot;
z^2_1 + z^2_2 = L^2 

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   <text:p text:style-name="Standard">The DAE in implicit form:</text:p>
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             <mrow>
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                <mo stretchy="false">˙</mo>
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              <msub>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
{dot z}_1 - z_3 =0
newline
&quot;&quot;
{dot z}_2 - z_4 =0
newline
&quot;&quot;
{dot z}_3 + z_1 z_5 = 0
newline
&quot;&quot;
{dot z}_4 + z_2 z_5 + g  = 0
newline
&quot;&quot;
z^2_1 + z^2_2 - L^2 = 0

</annotation>
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   <text:p text:style-name="Standard">Differentiate wrt t the implicit form:</text:p>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
{ddot z}_1 - {dot z}_3 =0
newline
&quot;&quot;
{ddot z}_2 - {dot z}_4 =0
newline
&quot;&quot;
{ddot z}_3 + {dot z}_1 z_5 + z_1 {dot z}_5 = 0
newline
&quot;&quot;
{ddot z}_4 + {dot z}_2 z_5 + z_2 {dot z}_5  = 0
newline
&quot;&quot;
2z_1{dot z}_1 + 2z_2{dot z}_2= 0

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   <text:p text:style-name="Standard">Differentiate wrt t second time:</text:p>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
{dddot z}_1 - {ddot z}_3 =0
newline
&quot;&quot;
{dddot z}_2 - {ddot z}_4 =0
newline
&quot;&quot;
{dddot z}_3 + {ddot z}_1 z_5 + 2{dot z}_1 {dot z}_5 + z_1 {ddot z}_5 = 0
newline
&quot;&quot;
{dddot z}_4 + {ddot z}_2 z_5 + 2{dot z}_2 {dot z}_5 + z_2 {ddot z}_5  = 0
newline
&quot;&quot;
2{dot z}_1{dot z}_1 +2z_1{ddot z}_1 + 2{dot z}_2{dot z}_2+ 2z_2{ddot z}_2= 0

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   <text:p text:style-name="Standard">Differentiate wrt t third time:</text:p>
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z^{&apos;&apos;&apos;&apos;}_1 - {dddot z}_3 =0
newline
&quot;&quot;
z^{&apos;&apos;&apos;&apos;}_2 - {dddot z}_4 =0
newline
&quot;&quot;
z^{&apos;&apos;&apos;&apos;}_3 + {dddot z}_1 z_5 + 3{ddot z}_1 {dot z}_5 + 3{dot z}_1 {ddot z}_5 + z_1 d{ddot z}_5 = 0
newline
&quot;&quot;
z^{&apos;&apos;&apos;&apos;}_4 + {dddot z}_2 z_5 + 3{ddot z}_2 {dot z}_5 + 3{dot z}_2 {ddot z}_5 + z_2 {dddot z}_5  = 0
newline
&quot;&quot;
6{dot z}_1{ddot z}_1 +2z_1{dddot z}_1 + 6{dot z}_2{ddot z}_2+ 2z_2{dddot z}_2= 0

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