| Summary: | In this paper, the pantograph delay differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mi>y</mi><mfenced separators="" open="(" close=")"><mi>c</mi><mi>t</mi></mfenced></mrow></semantics></math></inline-formula> subject to the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>λ</mi></mrow></semantics></math></inline-formula> is reanalyzed for the real constants <i>a</i>, <i>b</i>, and <i>c</i>. In the literature, it has been shown that the pantograph delay differential equation, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, is well-posed if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, but not if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. In addition, the solution is available in the form of a standard power series when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="|" close="|"><mfrac><mi>b</mi><mi>a</mi></mfrac></mfenced><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> and it also converges for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></msqrt></mfrac></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>></mo><mi>a</mi></mrow></semantics></math></inline-formula>. The current results are introduced for the first time and have not been reported in the relevant literature.
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